Application of Integration

Exercises 1.

2.

Sketch on the same diagram, the curves đ?‘Ś = đ?‘Ľ 2 + 1 and đ?‘Ś = −đ?‘Ľ 2 + 9. Hence, find a)

Points of intersection

b)

The area of the region bounded by these two curves.

Given đ?‘“(đ?‘Ľ) = { −đ?‘Ľ 3

1 1 − 2đ?‘Ľ+8 8

; â&#x2C6;&#x2019;4 < đ?&#x2018;Ľ â&#x2030;¤ 0

; 0<đ?&#x2018;Ľ<â&#x2C6;&#x17E;

Find the area of the region bounded by f(x). Leave your final answer correct to three decimal places.

3.

The volume of solid formed by revolving region A through 360° about y-axis is two times the volume of the solid formed by revolving region B through 360° about x-axis. Find the value of đ?&#x153;&#x2020;.

4.

Sketch the area of the region bounded by đ?&#x2018;Ś = 2 â&#x2C6;&#x2019; đ?&#x2018;Ľ 2 and đ?&#x2018;Ś = đ?&#x2018;Ľ . Hence, find the first moment of the region about y-axis.

5.

Sketch the area of the region bounded by đ?&#x2018;Ś = â&#x2C6;&#x2019;3đ?&#x2018;Ľ + 9 and đ?&#x2018;Ś = 9 â&#x2C6;&#x2019; đ?&#x2018;Ľ 2 . Hence, find the volume of solid generated by rotating the region about x-axis.

6.

The shaded region below is bounded by the curve y= 2x2, the y-axis and the line y = 4 - 2x.

y y=4-2x

y=2x2

x

Find a) the area of the shaded region, b) the volume of the solid generated by revolving the region about the x-axis. c) the first moment with respect to x-axis.

7.

The shaded region below is bounded by the curves y = e-2x, y = ex and the line x = 1.

y y=ex x=1 y=e-2x

x

a) Find the area of the shaded region. b) Find the volume of the solid generated by revolving the region about the x-axis. c) Set up the integrals to compute the first moment with respect to x-axis and y-axis. d) Locate the centroid if Qx ď&#x20AC;˝ 1.4745 and Qy ď&#x20AC;˝ 0.8515 .

8.

Refer to the figure given below.

a)

Find the area of the shaded region.

b)

Find the volume of the solid of revolution obtained when the region bounded by the curve y = 9 - x2 and the line y = -3x + 9 is revolved about the line x = 0.

c)

9.

Find the first moment with respect to x-axis.

The shaded region below is bounded by the curve y  x = 5.

x  1 , the x-axis and the line

y  x 1

a)

Find the area of the shaded region.

b) Find the volume of the solid generated when the region R, is revolved about y-axis. c)

Find the first moment with respect to y-axis.

10. a) Sketch the area of the region bounded by đ?&#x2018;Ś = 4 and đ?&#x2018;Ś = â&#x2C6;&#x161;đ?&#x2018;Ľ â&#x2C6;&#x2019; 2 and y-axis. b) Find the area of the shaded region by using horizontal sampling strip. c) Find the first moment of area about x-axis. d) Find the moment of inertia with respect to x-axis.

Answers :

1.

a) : đ?&#x2018;Ľ = 2 đ?&#x2018;&#x153;đ?&#x2018;&#x; â&#x2C6;&#x2019; 2

2.

1.068đ?&#x2018;˘đ?&#x2018;&#x203A;đ?&#x2018;&#x2013;đ?&#x2018;Ą 2

3.

đ?&#x153;&#x2020; = 2 đ?&#x2018;&#x2122;đ?&#x2018;&#x203A;40

4.

5 đ?&#x2018;˘đ?&#x2018;&#x203A;đ?&#x2018;&#x2013;đ?&#x2018;Ą 3 12

5.

243 đ?&#x153;&#x2039; 5

6.

a)

7.

a) 1.286

b) đ??´đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D; =

64 đ?&#x2018;˘đ?&#x2018;&#x203A;đ?&#x2018;&#x2013;đ?&#x2018;Ą 2 3

1

đ?&#x2018;˘đ?&#x2018;&#x203A;đ?&#x2018;&#x2013;đ?&#x2018;Ą 3

7 3

b)

ď&#x192;˛

e ď&#x20AC;­2

d)

c)

64 15

b) 2.949 ď ° 1

c) Q x ď&#x20AC;˝

128 ď ° 15

ď&#x192;Ś ď&#x192;Ś ln y ď&#x192;ś ď&#x192;ś y ď&#x192;§ď&#x192;§1 ď&#x20AC;Ť ď&#x192;§ ď&#x192;ˇ ď&#x192;ˇď&#x192;ˇdy ď&#x20AC;Ť ď&#x192;¨ ď&#x192;¨ 2 ď&#x192;¸ď&#x192;¸

centroid : ď&#x20AC;¨0.66,1.15 ď&#x20AC;Š

8.

a) 4.5

b)

27 ď ° 2

9.

a)

16 3

b)

544 ď ° 15

10.

b ) đ??´đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D; = 29.3 đ?&#x2018;˘đ?&#x2018;&#x203A;đ?&#x2018;&#x2013;đ?&#x2018;Ą 2

c)

c)

e

ď&#x192;˛ yď&#x20AC;¨1 ď&#x20AC;­ ln yď&#x20AC;Šdy

1

, Qy ď&#x20AC;˝

1

ď&#x192;˛ x ď&#x20AC;¨e

x

ď&#x20AC;Š

ď&#x20AC;­ e ď&#x20AC;­2 x dx

0

243 10 272 15 c) đ?&#x2018;&#x201E;đ?&#x2018;Ľ = 80 đ?&#x2018;˘đ?&#x2018;&#x203A;đ?&#x2018;&#x2013;đ?&#x2018;Ą 3

d) đ??źđ?&#x2018;Ľ = 247.47 đ?&#x2018;˘đ?&#x2018;&#x203A;đ?&#x2018;&#x2013;đ?&#x2018;Ą 4