Application of Integrals — Practice

Question 1

Find the area bounded by y=x^2, x=1, x=2, and the x-axis.

Accepted Answer

Since x^2 is positive throughout [1,2], the area is integral from 1 to 2 of x^2 dx = [x^3/3] from 1 to 2 = 8/3-1/3. Answer: 7/3 square units.

Question 2

Find the area bounded by y=x^4, x=1, x=5, and the x-axis.

Accepted Answer

Since x^4 is positive throughout [1,5], the area is integral from 1 to 5 of x^4 dx = [x^5/5] from 1 to 5 = 3125/5-1/5. Answer: 3124/5 square units.

Question 3

Find the area of the region bounded by the ellipse x^2/16+y^2/9=1.

Accepted Answer

Here a=4, b=3, so using Area=pi.a.b gives the result directly. Answer: 12*pi square units.

Question 4

Find the area lying in the first quadrant and bounded by the circle x^2+y^2=4 and the lines x=0 and x=2.

Accepted Answer

This region is exactly one quarter of the full circle of radius 2, since it is bounded away from the axes only by the arc between x=0 and x=2. Answer: pi square units (since a quarter of pi.r^2=4pi is pi).

Question 5

Find the area of the region bounded by the curve y^2=4x, the y-axis, and the line y=3.

Accepted Answer

Using the y-axis formula, Area = integral from 0 to 3 of x dy = integral from 0 to 3 of (y^2/4) dy = [y^3/12] from 0 to 3 = 27/12. Answer: 9/4 square units.

Question 6

Find the area of the region bounded by the line y=3x+2, the x-axis, and the ordinates x=-1 and x=1.

Accepted Answer

Since y=3x+2 crosses the x-axis at x=-2/3, split the integral there: |integral from -1 to -2/3 of (3x+2)dx| + integral from -2/3 to 1 of (3x+2)dx = 1/6 + 25/6. Answer: 13/3 square units.

Question 7

Find the area bounded by the curve y=sin x between x=0 and x=2*pi.

Accepted Answer

Since sinx is positive on [0,pi] and negative on [pi,2pi], split the integral there: integral from 0 to pi of sinx dx + |integral from pi to 2pi of sinx dx| = 2+2. Answer: 4 square units (not 0, which the naive unsplit integral would wrongly give).

Question 8

The area bounded by the curve y=x|x|, the x-axis, and the ordinates x=-1 and x=1 is given by which of

Accepted Answer

C

Practice — Application of Integrals

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