[WWW.toanPhoThong.tk] Bai Tap Ung Dung Cua Tich Phan

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<p>WWW.ToanPhoThong.TKCHUYN </p> <p>NG DNG CA TCH PHNA. TCH PHN CHA GI TR TUYT I Phng php gii ton 1. Dng 1b</p> <p>I =</p> <p>a Gi s cn tnh tch phn , ta thc hin cc bc sau Bc 1. Lp bng xt du (BXD) ca hm s f(x) trn on [a; b], gi s f(x) c BXD:</p> <p> f(x) dxx1 0</p> <p>x f(x)b</p> <p>a</p> <p>+x1</p> <p>-</p> <p>x2 0x2</p> <p>+</p> <p>b</p> <p>b</p> <p>I = Bc 2. Tnh</p> <p> f(x) dx = f(x)dx - f(x)dx + f(x)dxa a x1 x2 2</p> <p>.</p> <p>I = V d 1. Tnh tch phn Bng xt du1</p> <p>x- 3</p> <p>2</p> <p>- 3x + 2 dx . Gii - 3 +2</p> <p>x x - 3x + 22</p> <p>1 0</p> <p>-</p> <p>2 0 59 2 .</p> <p>I =</p> <p>( x- 3</p> <p>2</p> <p>- 3x + 2) dx p 2</p> <p>( x1</p> <p>2</p> <p>- 3x + 2) dx =</p> <p>I = V d 2. Tnh tch phnp 2</p> <p>0</p> <p>5 - 4cos2 x - 4sin xdx . Giip 2</p> <p>I = Bng xt du</p> <p>0</p> <p>4sin2 x - 4sin x + 1 dx = x 2sin x - 1 0 p 6 0</p> <p> 2sin x 0</p> <p>1 dx .</p> <p>+</p> <p>p 2</p> <p>WWW.ToanPhoThong.TK</p> <p>1</p> <p>WWW.ToanPhoThong.TKp 6 p 2</p> <p>I =-</p> <p> ( 2sin x 0</p> <p>1) dx + ( 2sin x - 1) dx = 2 3 - 2 p 6</p> <p>p 6 .</p> <p>2. Dng 2b</p> <p>I = Gi s cn tnh tch phn Cch 1.b</p> <p> [ f(x)a b</p> <p> g(x) ] dx , ta thc hin:b</p> <p>I = Tch</p> <p> [ f(x)a</p> <p> g(x) ] dx =</p> <p> f(x) dx g(x) dxa a</p> <p>ri s dng dng 1 trn.</p> <p>Cch 2. Bc 1. Lp bng xt du chung ca hm s f(x) v g(x) trn on [a; b]. Bc 2. Da vo bng xt du ta b gi tr tuyt i ca f(x) v g(x).2</p> <p>I = V d 3. Tnh tch phn Cch 1.2</p> <p>(- 1</p> <p>x - x - 1 ) dx . Gii2 2</p> <p>I = =-</p> <p>(- 1 0 - 1</p> <p>x - x - 1 ) dx =2 1</p> <p> x dx - x - 1 - 1 2</p> <p>1 dx 1 )dx</p> <p> xdx + xdx + (x 2 0 - 1</p> <p>1 )dx 1 2</p> <p>x =2 Cch 2. Bng xt du</p> <p>x + 2</p> <p>0 2 2</p> <p>- 1</p> <p> (x 1 2</p> <p>0</p> <p> x + - x 2 - 12</p> <p> x - x =0 2 1 . 2 + +2 1</p> <p>x x x10</p> <p>1 </p> <p>0 01</p> <p>1 + 0</p> <p>I =</p> <p>( - x + x - 1</p> <p>1) dx + ( x + x - 1) dx + ( x - x + 1) dx0 2 1 0 2 +x1 = 0.</p> <p>= - x -0 1 + ( x - x )</p> <p>Vy I = 0 .</p> <p>3. Dng 3</p> <p>WWW.ToanPhoThong.TK</p> <p>2</p> <p>WWW.ToanPhoThong.TKb b</p> <p>I = tnh cc tch phn thc hin cc bc sau:</p> <p> max { f(x),a</p> <p>g(x) } dx v</p> <p>J =</p> <p> min { f(x),a</p> <p>g(x) } dx , ta</p> <p>Bc 1. Lp bng xt du hm s h(x) = f(x) - g(x) trn on [a; b]. Bc 2. + Nu h(x) &gt; 0 th max { f(x), g(x) } = f(x) v min { f(x), g(x) } = g(x) . + Nu h(x) &lt; 0 th max { f(x), g(x) } = g(x) v min { f(x), g(x) } = f(x) .4</p> <p>I = V d 4. Tnh tch phn</p> <p> max { x0 2</p> <p>2</p> <p>+1 , 4x - 2} dx .</p> <p>Gii 2 ( ) t h(x) = ( x + 1) - 4x - 2 = x - 4x + 3 . Bng xt du x h(x)1</p> <p>0 +2</p> <p>1 03 1</p> <p>3 0</p> <p>4 +4 3</p> <p>I =</p> <p>( x0</p> <p>+ 1) dx + ( 4x - 2) dx + ( x2 + 1) dx =2</p> <p>80 3</p> <p>.</p> <p>I = V d 5. Tnh tch phn</p> <p> min { 3 ,x 0</p> <p>4 - x } dx .</p> <p>Gii x ( 4 - x ) = 3x + x - 4 h(x) = 3 t . Bng xt du x h(x)1 2</p> <p>0 </p> <p>1 0x</p> <p>2 +1 2</p> <p>I =</p> <p> 3x dx + ( 4 - x ) dx =0 1</p> <p> 3 x2 2 5 + 4x = + ln 3 0 2 ln 3 2 1</p> <p>.</p> <p>B. NG DNG CA TCH PHNI. DIN TCH HNH PHNG</p> <p>1. Din tch hnh thang cong</p> <p>WWW.ToanPhoThong.TK</p> <p>3</p> <p>WWW.ToanPhoThong.TKCho hm s f(x) lin tc trn on [a; b]. Din tch hnh thang cong gii hn bi ccb</p> <p>ng y = f(x), x = a, x = b v trc honh l:</p> <p>S = f(x) dxa</p> <p>.</p> <p>Phng php gii tonBc 1. Lp bng xt du hm s f(x) trn on [a; b].b</p> <p>Bc 2. Da vo bng xt du tnh tch phn</p> <p> f(x) dxa</p> <p>.</p> <p>, x = e v Ox. V d 1. Tnh din tch hnh phng gii hn bi y = ln x, x = 1 Gii [1 ] ln x 0 " x ; e Do nn:e e</p> <p>S=</p> <p> ln x dx = ln xdx = x ( ln x 1 1</p> <p>1)</p> <p>e 1</p> <p>Vy S = 1 (vdt).</p> <p>=1 .</p> <p>2 , x=0 , x = 3 v V d 2. Tnh din tch hnh phng gii hn bi y = - x + 4x - 3 Ox. Gii Bng xt du x 0 1 3 y 0 + 0 1 3 2</p> <p>S=-</p> <p>( - x0 3</p> <p>+ 4x - 3) dx + ( - x2 + 4x - 3) dx1 1 3</p> <p> x x3 8 2 =- + 2x + 3x + + 2x2 + 3x = 3 0 3 1 3. 8 S= 3 (vdt). Vy</p> <p>2. Din tch hnh phng 2.1. Trng hp 1Cho hai hm s f(x) v g(x) lin tc trn on [a; b]. Din tch hnh phng gii hn bib</p> <p>cc ng y = f(x), y = g(x), x = a, x = b l:</p> <p>S = f(x) - g(x) dxa</p> <p>.</p> <p>Phng php gii ton WWW.ToanPhoThong.TK 4</p> <p>WWW.ToanPhoThong.TKBc 1. Lp bng xt du hm s f(x) - g(x) trn on [a; b].b</p> <p>Bc 2. Da vo bng xt du tnh tch phn</p> <p> f(x) a</p> <p>g(x) dx .</p> <p>2.2. Trng hp 2Cho hai hm s f(x) v g(x) lin tc trn on [a; b]. Din tch hnh phng gii hn bib</p> <p>S = f(x) - g(x) dx y = f(x), y = g(x) a cc ng l: . a , b Trong l nghim nh nht v ln nht ca phng trnh f(x) = g(x)( a a &lt; b b) .</p> <p>Phng php gii tonBc 1. Gii phng trnh f(x) = g(x) . Bc 2. Lp bng xt du hm s f(x) - g(x) trn on [ a; b] .b</p> <p>Bc 3. Da vo bng xt du tnh tch phn</p> <p> f(x) a</p> <p>g(x) dx .</p> <p>V d 3. Tnh din tch hnh phng gii hn bi cc ng: y = x3 + 11x - 6 , y = 6x2 , x = 0 , x = 2. Gii 3 2 3 h(x) = (x + 11x 6) 6x = x 6x2 + 11x - 6 t h(x) = 0 x = 1 x = 2 x = 3 (loi). Bng xt du x 0 h(x)1</p> <p>2</p> <p>1 0</p> <p>+</p> <p>2 0</p> <p>S=-</p> <p>( x0 4</p> <p>3</p> <p>- 6x + 11x - 6) dx + ( x3 - 6x2 + 11x - 6) dx2 1 2 1 2</p> <p> x 11x x4 11x2 5 3 3 =- 2x + 6x + 2x + - 6x = 4 0 4 1 2 2 2. 5 S= 2 (vdt). Vy3 , y = 6x2 . V d 4. Tnh din tch hnh phng gii hn bi cc ng y = x + 11x - 6</p> <p>WWW.ToanPhoThong.TK</p> <p>5</p> <p>WWW.ToanPhoThong.TKGii 2 t h(x) = (x + 11x - 6) - 6x = x - 6x + 11x - 6 h(x) = 0 x = 1 x = 2 x = 3 .3 2 3</p> <p>Bng xt du x 1 h(x) 02 3 3</p> <p>+</p> <p>2 03</p> <p>3 0</p> <p>S=</p> <p>( x1 4</p> <p>- 6x + 11x - 6) dx 2 2 2</p> <p>( x2</p> <p>- 6x2 + 11x - 6) dx3</p> <p> x 11x = - 2x3 + - 6x 4 2 1 Vy S= 1 2 (vdt).</p> <p> x4 11x2 1 - 2x3 + - 6x = 4 2 2 2.</p> <p>Ch :1) Nu hnh phng c gii hn t 3 ng tr ln th phi v hnh, tuy nhin hu ht rt kh xc nh ng min phng cn tnh din tch (c th v th m thi i hc khng ra). 2) Nu trong khong dng cng thc:</p> <p>( a; b)b</p> <p>phng trnh f(x) = g(x) khng c nghim th ta c thb</p> <p> f(x) a</p> <p>g(x) dx =</p> <p>f(x) a</p> <p>g(x) dx </p> <p>3) Nu tch din tch hnh phng gii hn bi x = f(y) v x = g(y) th ta gii nh trn nhng nh i vai tr x cho y (xem v d 9).3 V d 5. Tnh din tch hnh phng gii hn bi y = x , y = 4x .</p> <p>Gii Phng trnh honh giao im: x3 = 4x x = - 2 x = 0 x = 20 2 3</p> <p> S=</p> <p>( x- 2</p> <p>- 4x ) dx +0</p> <p>( x0</p> <p>3</p> <p>- 4x ) dx</p> <p> x4 x4 2 = 2x + - 2x2 =8 4 - 2 4 0 Vy S = 8 (vdt).</p> <p>2</p> <p>.</p> <p>2 V d 6. Tnh din tch hnh phng gii hn bi y = x - 4 x + 3 v trc honh. Gii Phng trnh honh giao im:</p> <p>WWW.ToanPhoThong.TK</p> <p>6</p> <p>WWW.ToanPhoThong.TKx2 - 4 x + 3 = 0 t2 t =1 x = 1 x = 3 t=3 3</p> <p>- 4t + 3 = 0 , t= x 0 x = 1 x = 3 3 0</p> <p> S=</p> <p>x- 3</p> <p>2</p> <p>- 4 x + 3 dx = 2 x2 - 4x + 3 dx</p> <p>3 1 2 2 = 2 x 4x + 3 dx + x 4x + 3 dx ( ) ( ) 1 0 1 3 3 3 x x 16 = = 2 - 2x2 + 3x + - 2x2 + 3x 0 3 1 3 3</p> <p>16 S= 3 (vdt). Vy</p> <p>.</p> <p>2 V d 7. Tnh din tch hnh phng gii hn bi y = x - 4x + 3 v y = x + 3 . Gii Phng trnh honh giao im: x + 3 0 x=0 x2 - 4x + 3 = x + 3 x=5 2 2 x - 4x + 3 = - x - 3 x - 4x + 3 = x + 3 . Bng xt du x 0 1 3 5 + 0 0 + x2 - 4x + 3 1 3 2 2 5</p> <p> S=</p> <p>( x0 3</p> <p>- 5x ) dx + ( - x + 3x - 6) dx + ( x2 - 5x ) dx1 3 2 1 3 2 3 3 5</p> <p> x 5x -x 3x x 5x2 109 = + + 6x + = 3 3 3 2 2 2 6 0 1 3 109 S= 6 (vdt). Vy</p> <p>.</p> <p>2 V d 8. Tnh din tch hnh phng gii hn bi y = x - 1 , y = x + 5 . Gii Phng trnh honh giao im: x2 - 1 = x + 5 t2 - 1 = t + 5 , t= x 0 t= x 0 t= x 0 t2 - 1 = t + 5 x = 3 t=3 2 t - 1= - t - 5 3 3</p> <p> S=</p> <p>- 3</p> <p>x2 - 1 -</p> <p>(</p> <p>x + 5) dx = 2 x2 - 1 0</p> <p>(</p> <p>x + 5) dx</p> <p>WWW.ToanPhoThong.TK</p> <p>7</p> <p>WWW.ToanPhoThong.TKBng xt du x x - 12 1 3 2</p> <p>0 1</p> <p>1 0</p> <p>3 +</p> <p> S=2</p> <p>( - x0 3</p> <p>- x - 4) dx + ( x2 - x - 6) dx2 1 3</p> <p> -x x x3 x2 73 =2 4x + 6x = 3 3 2 2 3 0 1 73 S= 3 (vdt). Vy</p> <p>.</p> <p>, y= V d 9. Tnh din tch hnh phng gii hn bi y = x, y = 0 Gii 2 2 Ta c: y = 2 - x x = 2 - y , x 0. Phng trnh tung giao im: y =1</p> <p>2 - x2 .</p> <p>2 - y2 y = 1. 2 - y2 - y ) dy</p> <p> S=</p> <p>2 - y - y dy =1</p> <p>2</p> <p>1 = 2cos tdt - ydy = t + sin2t 2 0 02</p> <p>0 p 4</p> <p>(0</p> <p>1</p> <p>(</p> <p>)</p> <p>p 4 0</p> <p>y2 2</p> <p>1</p> <p>0</p> <p>p S= 4 (vdt). Vy Cch khc:</p> <p>.</p> <p>1 V hnh ta thy S bng 8 din tch hnh trn bn knh R =</p> <p>2 nn</p> <p>S=</p> <p>1 2 p pR = 8 4.</p> <p>II. TH TCH KHI TRN XOAY 1. Trng hp 1Th tch khi trn xoay do hnh phng gii hn bi cc ng y = f(x) 0" x [ a;b ] ,b</p> <p>y = 0 , x = a v x = b (a &lt; b) quay quanh trc Ox l: a . 2 2 2 V d 1. Tnh th tch hnh cu do hnh trn (C) : x + y = R quay quanh Ox. Gii 2 x = R 2 x = R . Honh giao im ca (C) v Ox l 2 2 2 2 2 2 Phng trnh (C) : x + y = R y = R - x x3 4pR 3 V = p ( R - x ) dx = 2p ( R 2 - x2 ) dx = 2p R 2x = 3 3 . - R 0 02 2 R R R</p> <p>V = p f 2(x)dx</p> <p>WWW.ToanPhoThong.TK</p> <p>8</p> <p>WWW.ToanPhoThong.TKV = 4pR 3 3 (vtt).</p> <p>Vy</p> <p>2. Trng hp 2</p> <p>Th tch khi trn xoay do hnh phng gii hn bi cc ng x = g(y) 0" y [ c;d ] ,d c x = 0 , y = c v y = d (c &lt; d) quay quanh trc Oy l: . 2 2 x y (E) : 2 + 2 = 1 a b V d 2. Tnh th tch hnh khi do ellipse quay quanh Oy. Gii y2 = 1 y = b 2 Tung giao im ca (E) v Oy l b . 2 2 2 2 x y ay (E) : 2 + 2 = 1 x2 = a2 a b b2 Phng trnh R 2 a2y2 2 a2y2 2 a2y3 4pa2b V = p a dy = 2 p a dy = 2 p a y = 2 2 b b 3 . 3b2 - b 0 0 b b</p> <p>V = p g2(y)dy</p> <p>Vy</p> <p>V =</p> <p>4pa2b 3 (vtt).</p> <p>3. Trng hp 3</p> <p>Th tch khi trn xoay do hnh phng gii hn bi cc ng y = f(x), y = g(x) , ,g(x) 0 " x [ a; b ]) quay quanh trc Ox l: x = a v x = b (a &lt; b, f(x) 0b</p> <p>V = p f 2(x) - g2(x) dxa</p> <p>.</p> <p>V d 3. Tnh th tch hnh khi do hnh phng gii hn bi cc ng y = x2, y2 = x quay quanh Ox. Gii x=0 x 0 4 x =x x =1 Honh giao im: .1</p> <p> V = p x - x dx = p4 0</p> <p> ( x4 - x ) dx = p 1 x5 - 1 x20</p> <p>1</p> <p>(5</p> <p>2</p> <p>)</p> <p>1</p> <p>=0</p> <p>Vy</p> <p>V =</p> <p>3p 10 (vtt).</p> <p>3p 10 .</p> <p>4. Trng hp 4</p> <p>WWW.ToanPhoThong.TK</p> <p>9</p> <p>WWW.ToanPhoThong.TKTh tch khi trn xoay do hnh phng gii hn bi cc ng x = f(y), x = g(y) , y = c v y = d (c &lt; d, f(y) 0 ,g(y) 0 " y [ c; d ]) quay quanh trc Oy l:d</p> <p>V = p f 2(y) - g2(y) dyc</p> <p>.</p> <p>2 V d 4. Tnh th tch hnh khi do hnh phng gii hn bi cc ng x = - y + 5 , x = 3 - y quay quanh Oy.</p> <p>Gii y=- 1 - y2 + 5 = 3 - y y=2 Tung giao im: .2 2 2 2 2</p> <p> V = p ( - y + 5) - ( 3 - y ) dy = p- 1 5 3 2</p> <p>( y- 1</p> <p>4</p> <p>- 11y2 + 6y + 16) dy</p> <p> y 11y 153p =p + 3y2 + 16y = 5 3 5 - 1 153p V = 5 (vtt). Vy</p> <p>.</p> <p>BI TPBi 1. Tnh din tch hnh phng gii hn bi cc ng c phng trnh sau , x = 2p 1) y = sin x, y = 0, x = 03 , x=2 2) y = x , y = 0 , x = - 1 2 2 3) y = x - 2x, y = - x + 4x 3 , x=2 4) y = x , y = 4x , x = - 1 2 , y = - 6x , x = 0 , x =1 5) y = - x - 5 2 , y = - 3x , x = 0 , x=2 6) y = - x - 2 2 7) y = - x - 2x, y = - x - 2 3 2 8) y = x - 2x - x + 2 v trc honh 2 3 9) y = x - 2x - x + 2 v trc honh</p> <p>10)</p> <p>y=</p> <p>x2 x2 4, y= 4 4 2</p> <p>2 2 11) y = - 4 - x , x + 3y = 0 2 12) y = x - 4x + 3 , y = 3 2 13) y = x - 4 x + 3 , y = 0</p> <p>WWW.ToanPhoThong.TK</p> <p>10</p> <p>WWW.ToanPhoThong.TKx = y, x = 3 2 ( y 0)</p> <p>4 - y2 2 1 x = 2, x = , y= 2 y 8 y 15) 14) 16) 17) 18) 19)</p> <p>, y = ln x , x = 2 , x=e 20) y = 0 1 1 p p y= ,y= x= , x= 2 2 6 3 sin x cos x , 21) 2 2 22) y = x , y = 4x , y = 4</p> <p>p 3p x = , x = y = (2 + cosx) sin x, y = 0 , 2 2 y = x 1 + x2 , y = 0 , x = 1 ln x y= ,y=0 x =1 , x=e 2 x , 1 + ln x y= ,y=0 x =1 , x=e x ,</p> <p>)(x - 2), y = 0 , x = - 2 , x=2 23) y = x(x + 1 x , x=2 24) y = xe , y = 0 , x = - 12 25) y = 4x, x - y + 1 = 0 , y = 0 3 , x + y - 1= 0 , y=0 26) x - y + 1 = 0</p> <p>Bi 2. Tnh th tch do hnh phng gii hn bi cc ng , x = 1 quay quanh Ox 1) y = 3x, y = x , x = 0 x2 , y = 2 y = 4, x = 0 2 2) , quay quanh Oy 2 3 ) , x = 2 v y = 0 quay quanh Ox 3) y = (x - 1 2 4) y = 4 - x, x = 0 quay quanh Oy y=2 2 5) (C) : x + (y - 4) = 4 quay quanh Oy x2 y2 (E) : + =1 16 9 6) ellipse quay quanh Ox x2 x2 (E) : + =1 16 9 7) ellipse quay quanh Oy 2 2 , y = 4 - x quay quanh Ox 8) y = x + 2 2 9) y = x , y = x quay quanh Ox 2 2 10) y = - 4 - x , x + 3y = 0 quay quanh Ox</p> <p>HNG DN GII</p> <p>WWW.ToanPhoThong.TK</p> <p>11</p> <p>WWW.ToanPhoThong.TKBi 1.2p p 2p</p> <p>S= 1) (vdt). S=</p> <p> sin x dx = sin xdx0 0 2 0 2</p> <p>+</p> <p> sin xdxp</p> <p>2p = - cosx p 0 + - cosx p =4</p> <p>- 1 - 1 2) 2 2 3) x - 2x = - x + 4x x = 0 x = 3 3</p> <p>x dx =</p> <p>3</p> <p> x dx +3 2</p> <p>x4 3 x dx = 4 03</p> <p>0</p> <p>+- 1</p> <p>x4 4</p> <p>2</p> <p>=0</p> <p>17 4</p> <p>(vdt).3</p> <p> S=</p> <p>0 2</p> <p> 2x3 - 3x2 (x - 2x) - (- x + 4x) dx = (2x - 6x)dx = 3 0 02 2</p> <p>=</p> <p>9(vdt). 3 4) x - 4x = 0 x = 0 x = 2 x = - 2 (loi).0 3 2 3</p> <p> S=4</p> <p>x- 1</p> <p>- 4x dx =0 4</p> <p> (x- 1</p> <p>- 4x)dx +2</p> <p> (x0</p> <p>3</p> <p>- 4x)dx</p> <p>. 23 S= 4 (vdt). Vy 2 5) x - 6x + 5 = 0 x = 1 x = 5 (loi).1</p> <p> x x = - 2x2 + - 2x2 4 - 1 4 0</p> <p> S=</p> <p>0</p> <p>x - 6x + 5 dx =2</p> <p> x3 2 (x 6x + 5)dx = - 3x2 + 5x 3 0 0</p> <p>1</p> <p>1</p> <p>7 S= 3 (vdt). Vy 2 6) x - 3x + 2 = 0 x = 1 x = 2 .2 1 2 2 2</p> <p>.</p> <p> S=</p> <p>x0 3</p> <p>- 3x + 2 dx =2 1</p> <p> (x0 3</p> <p>- 3x + 2)dx +2 2</p> <p> (x1</p> <p>2</p> <p>- 3x + 2)dx</p> <p> x 3x x 3x = + 2x + + 2x 3 0 3 2 2 12 7) - x - 2x = - x - 2 x = - 2 x = 1. 1 1</p> <p>= 1(vdt).1</p> <p> S=</p> <p>- 2</p> <p>x + x - 2 dx =2</p> <p> x3 x2 2 (x + x 2)dx = + - 2x 3 2 - 2 - 2</p> <p>9 S= 2 (vdt). Vy 3 2 8) x - 2x - x + 2 = 0 x = 2 x = 1.</p> <p>.</p> <p>WWW.ToanPhoThong.TK</p> <p>12</p> <p>WWW.ToanPhoThong.TK2</p> <p> S=1</p> <p>x- 1 3</p> <p>3</p> <p>- 2x2 - x + 2 dx2 2</p> <p>=</p> <p> (x- 1</p> <p>- 2x - x + 2)dx +4 3 2 1</p> <p> (x1</p> <p>3</p> <p>- 2x2 - x + 2)dx2</p> <p> x 2x x x4 2x3 x2 = + 2x + + 2x 4 - 1 4 3 2 3 2 1 Vy S= 37 12 (vdt).</p> <p>.</p> <p> t= x 0 2 3 x - 2x - x + 2 = 0 3 2 t 2t t + 2 = 0 9)2 2</p> <p>t= x 0 t =1 t=2 </p> <p>x = 1 x = 2 </p> <p>.</p> <p> S=</p> <p>- 2 1 0</p> <p>x 3 - 2x2 - x + 2 dx = 2 x3 - 2x2 - x + 2 dx0 2 3</p> <p>=2</p> <p> (x4</p> <p>- 2x2 - x + 2)dx + 23 2 1</p> <p> (x1</p> <p>3</p> <p>- 2x2 - x + 2)dx2</p> <p> x 2x x x4 2x3 x2 =2 + 2x + 2 + 2x 4 4 3 2 3 2 0 1 10) 4x x = x4 + 8x2 - 128 = 0 x = 2 2 4 4 22 2 2 2</p> <p>= 3(vdt).</p> <p> S=</p> <p>- 2 2 2 2</p> <p>x2 x2 4dx = 4 4 2</p> <p> x2 x2 4dx 4 4 2 - 2 22 2</p> <p>2 2</p> <p> x2 x2 = 2 4dx = 4 4 2 0p 4 2 2 2</p> <p>0</p> <p>1 16 - x dx x2dx 2 2 02</p> <p>2 2</p> <p>1 = 16 cos tdt x2dx = 8 t + 1 sin2t 2 2 0 0 2</p> <p>(</p> <p>)</p> <p>p 4 0</p> <p>-</p> <p>1 x3 2 23</p> <p>2 2</p> <p>0</p> <p>4 S = 2p + 3 (vdt). Vy x2 x2 - 4 - x2 = 3 3 11) 4 2 x + 9x - 36 = 0 x = 3 x2 + 3y = 0 y = 3</p> <p>.</p> <p> S=</p> <p>-</p> <p>3</p> <p>x2 4- x dx = 2 32</p> <p> x2 2 4 x dx 3 0</p> <p>3</p> <p>WWW.ToanPhoThong.TK</p> <p>13</p> <p>WWW.ToanPhoThong.TK3</p> <p>=2</p> <p>0</p> <p>1 1 4 - x dx - x2dx = 2 4 cos2 tdt - x2dx 30 30 02</p> <p>3</p> <p>p 3</p> <p>3</p> <p>1 = 2 2 t + sin2t 2</p> <p>(</p> <p>)</p> <p>p 3 0</p> <p>-</p> <p>x3 9</p> <p>3</p> <p>0</p> <p>.</p> <p>4p + 3 S= 3 Vy (vdt). x2 - 4x + 3 = 3 x - 4x + 3 = 3 x2 - 4x + 3 = - 3 12) Bng xt du x 0 1 2 + 0 x - 4x + 32 4</p> <p>x=0 x=4 . 3 0 4 +</p> <p> S=1</p> <p>0 2</p> <p>x2 - 4x + 3 - 3 dx3 4 2</p> <p>=</p> <p>( x0</p> <p>- 4x ) dx +1</p> <p>( - x1</p> <p>+ 4x - 6) dx +3</p> <p>( x3</p> <p>2</p> <p>- 4x ) dx4</p> <p>= 8(vdt). x = 1 x = 1 x = 3 2 2 x = 3 13) x - 4 x + 3 = 0 x - 4 x + 3 = 0 . Bng xt du x 0 1 3 2 + 0 0 x - 4x + 33 3 2</p> <p> x3 - x3 x3 2 = - 2x2 + + 2x 6x + - 2x2 3 0 3 1 3 3</p> <p> S=</p> <p>x- 3</p> <p>- 4 x + 3 dx = 2 x2 - 4x + 3 dx0</p> <p>3 1 2 2 = 2 x 4x + 3 dx x 4x + 3 dx ( ) ( ) 1 0 1 3 3 3 x x = 2 - 2x2 + 3x - - 2x2 + 3x 3 3 0 1 . 16 S= 3 (vdt). Vy y=1 3 y= , 0 y 0 "x [1 ; e] 2 x 2 x 2 x 1 1 18) . t t t t = ln x x = e dx = e dt x = 1 t = 0 , x = e t = 1</p> <p>(</p> <p>)</p> <p>tet dt S= = t 0 2 e Vy S = 2 e</p> <p>1</p> <p>1</p> <p> td(0</p> <p>e)=t et e</p> <p>t 1 0</p> <p>1</p> <p>-</p> <p>0</p> <p>et dt =</p> <p>e - 2 et</p> <p>1 0</p> <p>S= 19) t</p> <p>1</p> <p>e (vdt). 1 + ln x dx = x</p> <p>.</p> <p>1</p> <p>1 + ln x dx x</p> <p>.</p> <p>t = 1 + ln x t2 = 1 + ln x 2tdt = x = 1 t = 1 , x = e t = 2</p> <p>dx x</p> <p>WWW.ToanPhoThong.TK</p> <p>15</p> <p>WWW.ToanPhoThong.TK2 S = t.2tdt = 2t dt = t3 3 1 12 2 2 2 1</p> <p>.</p> <p>4 2- 2 S= 3 Vy (vdt).e e</p> <p>e e 2</p> <p>S=</p> <p>20) Vy S = 2 - 2ln2 . 1 1 p p p = x= ; 2 2 4 6 3 sin x 21) cos x2 2</p> <p> ln x dx...</p>