Question 1

Passage: Let k be the area between the curve y= x and x-axis in the interval [0, 4 ]. Question: What is the area between the curve y= x and the x-axis in the interval [ 4 , 2 ] ?

Accepted Answer

Given k = _ 0 ^ /4 x \, dx k = [- x]_ 0 ^ /4 = - ( 4 ) + (0) = 1 - 1 2 The required area is A = _ /4 ^ /2 x \, dx A = [- x]_ /4 ^ /2 = - ( 2 ) + ( 4 ) = 1 2 From the expression for k, we have 1 2 = 1 - k Therefore, A = 1 - k Answer: 1-k

Question 2

Passage: Consider the curves y=x^2 and y=2|x|. Question: What is the area bounded by the curves, the lines x=0 and x=1 ?

Accepted Answer

For x [0, 1], the curve y = 2|x| becomes y = 2x. In the interval x [0, 1], 2x x^2. The required area is given by the integral: A = _ 0 ^ 1 (2x - x^2) dx A = [ x^2 - x^3 3 ]_ 0 ^ 1 A = 1 - 1 3 = 2 3 square units.

Question 3

Passage: Let k be the area between the curve y= x and x-axis in the interval [0, 4 ]. Question: What is the area between the curve y= x and the x-axis in the interval [ 4 , 2 ] ?

Accepted Answer

k = _ 0 ^ /4 x \, dx Let the required area be A = _ /4 ^ /2 x \, dx Substituting x = 2 - t, we get dx = -dt When x = 4 , t = 4 and when x = 2 , t = 0 A = _ /4 ^ 0 ( 2 - t ) (-dt) A = _ 0 ^ /4 t \, dt A = k Answer: k

Question 4

Consider the following for the two (02) items that follow: Let the curve f(x) = |x - 3|. What is the area bounded by the curve f(x) and y = 3?

Accepted Answer

The given curve is y = |x - 3| and the line is y = 3. To find the points of intersection, we equate the two equations: |x - 3| = 3 x - 3 = 3 or x - 3 = -3 x = 6 or x = 0 The points of intersection are (0, 3) and (6, 3). The vertex of the curve y = |x - 3| is at (3, 0). The area bounded by the curve and the line forms a triangle with vertices (0, 3), (6, 3), and (3, 0). The length of the base of this triangle (along the line y = 3) is 6 - 0 = 6 units. The height of the triangle (perpendicular dis

Question 5

Consider the following for the two (02) items that follow: Let the function f(x) = x^2 - 1. What is the area bounded by the function f(x) and the x-axis?

Accepted Answer

The curve intersects the x-axis when f(x) = 0. x^2 - 1 = 0 x = -1, 1 The required area is given by A = _ -1 ^ 1 |x^2 - 1| dx. Since x^2 - 1 0 for x [-1, 1], we have: A = _ -1 ^ 1 (1 - x^2) dx A = [ x - x^3 3 ]_ -1 ^ 1 A = ( 1 - 1 3 ) - ( -1 + 1 3 ) A = 2 3 - ( - 2 3 ) = 4 3 square units. Answer: 4 3 square units

Question 6

For the following three (03) items: Consider the function f(x) = x|x|. What is the area bounded by the curve f(x), the x-axis and the lines x = -2 and x = 1?

Accepted Answer

The given function is f(x) = x|x|. The required area bounded by the curve f(x), the x-axis and the lines x = -2 and x = 1 is given by: A = _ -2 ^ 1 |f(x)| dx A = _ -2 ^ 1 |x|x|| dx Since |x|x|| = |x|^2 = x^2, we have: A = _ -2 ^ 1 x^2 dx A = [ x^3 3 ]_ -2 ^ 1 A = 1^3 3 - (-2)^3 3 A = 1 3 - ( - 8 3 ) A = 9 3 = 3 Answer: 3

Question 7

Consider the following for the two (02) items that follow: The slope of the tangent to the curve y = f(x) at (x, f(x)) is 4 for every real number x and the curve passes through the origin. What is the

Accepted Answer

Given dy dx = 4 Integrating both sides, y = 4x + C Since the curve passes through the origin (0,0), we have 0 = 4(0) + C C = 0 The equation of the curve is y = 4x The area bounded by the curve, the x-axis and the line x = 4 is given by _ 0 ^ 4 y \, dx Area = _ 0 ^ 4 4x \, dx = [ 2x^2 ]_ 0 ^ 4 = 2(16) - 0 = 32 square units. Answer: 32 square units

Question 8

What is the area of the region bounded by |x| 2k and |y| k, where k is a positive real number?

Accepted Answer

The given inequalities are |x| 2k and |y| k. This can be written as -2k x 2k and -k y k. These inequalities represent a rectangular region in the xy-plane. The length of the rectangle along the x-axis is 2k - (-2k) = 4k. The width of the rectangle along the y-axis is k - (-k) = 2k. The area of the rectangular region is given by length width. Area = 4k 2k = 8k^2. Answer: 8k^2

Question 9

Consider the following for the two (02) items that follow: Let f(x) = [x^2], where [ ] is the greatest integer function. What is _ 2 ^ 3 f(x) \, dx equal to?

Accepted Answer

Given f(x) = [x^2]. The limits of integration are from x = 2 to x = 3 . For x ( 2 , 3 ), the value of x^2 lies in the interval (2, 3). Therefore, the greatest integer function [x^2] = 2. The integral is given by: _ 2 ^ 3 f(x) \, dx = _ 2 ^ 3 [x^2] \, dx = _ 2 ^ 3 2 \, dx = 2[x]_ 2 ^ 3 = 2( 3 - 2 )

Area Under Curves — Previous Year Questions & Solutions