JEE Mathematics Area Under The Curve MCQs Set 01

Question. The area of the region bounded by the curves \( y = | x - 2 | \), \( x = 1 \), \( x = 3 \) and the x-axis is
(a) 3
(b) 2
(c) 1
(d) 4
Answer: (c) 1

Question. The area enclosed between the curve \( y = \log_e(x + e) \) and the coordinate axes is
(a) 4
(b) 3
(c) 2
(d) 1
Answer: (d) 1

Question. The area of the figure bounded by the curves \( y = \ln x \) & \( y = (\ln x)^2 \) is
(a) \( e + 1 \)
(b) \( e - 1 \)
(c) \( 3 - e \)
(d) 1
Answer: (c) \( 3 - e \)

Question. The area enclosed by the curves \( y = \cos x \), \( y = 1 + \sin 2x \) and \( x = \frac{3\pi}{2} \) as \( x \) varies from 0 to \( \frac{3\pi}{2} \), is
(a) \( \frac{3\pi}{2} - 2 \)
(b) \( \frac{3\pi}{2} \)
(c) \( 2 + \frac{3\pi}{2} \)
(d) \( 1 + \frac{3\pi}{2} \)
Answer: (c) \( 2 + \frac{3\pi}{2} \)

Question. Let 'a' be a positive constant number. Consider two curves \( C_1 : y = e^x \), \( C_2 : y = e^{a - x} \). Let S be the area of the part surrounding by \( C_1 \), \( C_2 \) and the y-axis, then \( \lim_{a \to 0} \frac{S}{a^2} \) equals
(a) 4
(b) 1/2
(c) 0
(d) 1/4
Answer: (d) 1/4

Question. Suppose \( y = f(x) \) and \( y = g(x) \) are two functions whose graphs intersect at the three points (0, 4), (2, 2) and (4, 0) with \( f(x) > g(x) \) for \( 0 < x < 2 \) and \( f(x) < g(x) \) for \( 2 < x < 4 \).
If \( \int_0^4 [f(x) - g(x)] \, dx = 10 \) and \( \int_2^4 [g(x) - f(x)] \, dx = 5 \), then area between two curves for \( 0 < x < 2 \), is
(a) 5
(b) 10
(c) 15
(d) 20
Answer: (c) 15

Question. The area enclosed by the curve \( y^2 + x^4 = x^2 \) is
(a) \( \frac{2}{3} \)
(b) \( \frac{4}{3} \)
(c) \( \frac{8}{3} \)
(d) \( \frac{10}{3} \)
Answer: (b) \( \frac{4}{3} \)

Question. The area of the region (s) enclosed by the curves \( y = x^2 \) and \( y = \sqrt{| x |} \) is
(a) 1/3
(b) 2/3
(c) 1/6
(d) 1
Answer: (b) 2/3

Question. The area of the closed figure bounded by \( y = x \), \( y = -x \) & the tangent to the curve \( y = \sqrt{x^2 - 5} \) at the point (3, 2) is
(a) 5
(b) \( 2\sqrt{5} \)
(c) 10
(d) \( \frac{5}{2} \)
Answer: (a) 5

Question. The area bounded by the curve \( y = xe^{-x} \) ; \( xy = 0 \) and \( x = c \) where c is the x-coordinate of the curve's inflection point, is
(a) \( 1 - 3e^{-2} \)
(b) \( 1 - 2e^{-2} \)
(c) \( 1 - e^{-2} \)
(d) 1
Answer: (a) \( 1 - 3e^{-2} \)

Question. The line \( y = mx \) bisects the area enclosed by the curve \( y = 1 + 4x - x^2 \) & the line \( x = 0 \), \( x = \frac{3}{2} \) & \( y = 0 \). Then the value of m is
(a) \( \frac{13}{6} \)
(b) \( \frac{6}{13} \)
(c) \( \frac{3}{2} \)
(d) 4
Answer: (a) \( \frac{13}{6} \)

Question. The area bounded by the curves \( y = -\sqrt{-x} \) and \( x = -\sqrt{-y} \) where \( x, y \le 0 \)
(a) cannot be determined
(b) is 1/3
(c) is 2/3
(d) is same as that of the figure bounded by the curves \( y = \sqrt{-x} \) ; \( x \le 0 \) and \( x = \sqrt{-y} \) ; \( y \le 0 \)
Answer: (b) is 1/3

Question. If (a, 0) ; \( a > 0 \) is the point where the curve \( y = \sin 2x - \sqrt{3} \sin x \) cuts the x-axis first, A is the area bounded by this part of the curve, the origin and the positive x-axis, then
(a) \( 4A + 8 \cos a = 7 \)
(b) \( 4A + 8 \sin a = 7 \)
(c) \( 4A - 8 \sin a = 7 \)
(d) \( 4A - 8 \cos a = 7 \)
Answer: (a) \( 4A + 8 \cos a = 7 \)

Question. Consider two curves \( C_1 : y = \frac{1}{x} \) and \( C_2 : y = \ln x \) on the xy plane. Let \( D_1 \) denotes the region surrounded by \( C_1 \), \( C_2 \) and the line \( x = 1 \) and \( D_2 \) denotes the region surrounded by \( C_1 \), \( C_2 \) and the line \( x = a \). If \( D_1 = D_2 \) then the value of 'a'
(a) \( \frac{e}{2} \)
(b) \( e \)
(c) \( e - 1 \)
(d) \( 2(e - 1) \)
Answer: (b) \( e \)

Question. The area bounded by the curve \( y = f(x) \), the x-axis & the ordinates \( x = 1 \) & \( x = b \) is \( (b - 1) \sin (3b + 4) \). Then \( f(x) \) is
(a) \( (x - 1) \cos (3x + 4) \)
(b) \( \sin (3x + 4) \)
(c) \( \sin (3x + 4) + 3(x - 1) \cos (3x + 4) \)
(d) None of the options
Answer: (c) \( \sin (3x + 4) + 3(x - 1) \cos (3x + 4) \)

Question. The area of the region for which \( 0 < y < 3 - 2x - x^2 \) & \( x > 0 \) is
(a) \( \int_1^3 (3 - 2x - x^2) \, dx \)
(b) \( \int_0^3 (3 - 2x - x^2) \, dx \)
(c) \( \int_0^1 (3 - 2x - x^2) \, dx \)
(d) \( \int_{-1}^3 (3 - 2x - x^2) \, dx \)
Answer: (c) \( \int_0^1 (3 - 2x - x^2) \, dx \)

Question. The area bounded by the curves \( y = x(1 - \ln x) \) ; \( x = e^{-1} \) and positive x-axis between \( x = e^{-1} \) and \( x = e \) is
(a) \( \frac{e^2 - 4e^{-2}}{5} \)
(b) \( \frac{e^2 - 5e^{-2}}{4} \)
(c) \( \frac{4e^2 - e^{-2}}{5} \)
(d) \( \frac{5e^2 - e^{-2}}{4} \)
Answer: (b) \( \frac{e^2 - 5e^{-2}}{4} \)

Question. The curve \( f(x) = Ax^2 + Bx + C \) passes through the point (1, 3) and line \( 4x + y = 8 \) is tangent to it at the point (2, 0). The area enclosed by \( y = f(x) \), the tangent line and the y-axis is
(a) 4/3
(b) 8/3
(c) 16/3
(d) 32/3
Answer: (b) 8/3

Question. Let \( y = g(x) \) be the inverse of a bijective mapping \( f : \mathbb{R} \to \mathbb{R} \), \( f(x) = 3x^3 + 2x \). The area bounded by graph of \( g(x) \), the x-axis and the ordinate at \( x = 5 \) is
(a) 5/4
(b) 7/4
(c) 9/4
(d) 13/4
Answer: (d) 13/4

Question. A function \( y = f(x) \) satisfies the differential equation, \( \frac{dy}{dx} - y = \cos x - \sin x \), with initial condition that y is bounded when \( x \to \infty \). The area enclosed by \( y = f(x) \), \( y = \cos x \) and the y-axis in the 1st quadrant
(a) \( \sqrt{2} - 1 \)
(b) \( \sqrt{2} \)
(c) 1
(d) \( 1/\sqrt{2} \)
Answer: (a) \( \sqrt{2} - 1 \)

Question. If \( C_1 \equiv y = \frac{1}{1+x^2} \) and \( C_2 \equiv y = \frac{x^2}{2} \) be two curve lying in XY plane. Then
(a) area bounded by curve \( C_1 \) and y = 0 is \( \pi \)
(b) area bounded by \( C_1 \) and \( C_2 \) is \( \frac{\pi}{2} - \frac{1}{3} \)
(c) area bounded by \( C_1 \) and \( C_2 \) is \( 1 - \frac{\pi}{2} \)
(d) area bounded by curve \( C_1 \) and x-axis is \( \frac{\pi}{2} \)
Answer: (a) area bounded by curve \( C_1 \) and y = 0 is \( \pi \) and (b) area bounded by \( C_1 \) and \( C_2 \) is \( \frac{\pi}{2} - \frac{1}{3} \)

Question. Area enclosed by the curves \( y = \ln x \), \( y = \ln |x| \) ; \( y = |\ln x| \) and \( y = |\ln |x|| \) is equal to
(a) 2
(b) 4
(c) 8
(d) cannot be determined
Answer: (b) 4

Question. \( y = f(x) \) is a function which satisfies
(i) \( f(0) = 0 \) \quad (ii) \( f''(x) = f'(x) \) and (iii) \( f'(0) = 1 \)
then the area bounded by the graph of \( y = f(x) \), the lines \( x = 0 \), \( x - 1 = 0 \) and \( y + 1 = 0 \), is
(a) e
(b) e - 2
(c) e - 1
(d) e + 1
Answer: (c) e - 1

Question. Let T be the triangle with vertices (0, 0), \( (0, c^2) \) and \( (c, c^2) \) and let R be the region between \( y = cx \) and \( y = x^2 \) where \( c > 0 \) then
(a) Area (R) = \( \frac{c^3}{6} \)
(b) Area of R = \( \frac{c^3}{3} \)
(c) \( \lim_{c \to 0^+} \frac{\text{Area(T)}}{\text{Area(R)}} = 3 \)
(d) \( \lim_{c \to 0^+} \frac{\text{Area(T)}}{\text{Area(R)}} = \frac{3}{2} \)
Answer: (a) Area (R) = \( \frac{c^3}{6} \) and (c) \( \lim_{c \to 0^+} \frac{\text{Area(T)}}{\text{Area(R)}} = 3 \)

Question. Suppose \( g(x) = 2x + 1 \) and \( h(x) = 4x^2 + 4x + 5 \) and \( h(x) = (f \circ g)(x) \). The area enclosed by the graph of the function \( y = f(x) \) and the pair of tangents drawn to it from the origin, is
(a) \( \frac{8}{3} \)
(b) \( \frac{16}{3} \)
(c) \( \frac{32}{3} \)
(d) None of the options
Answer: (b) \( \frac{16}{3} \)

Question. Let \( f(x) = x^2 + 6x + 1 \) and R denote the set of points (x, y) in the coordinate plane such that \( f(x) + f(y) \le 0 \) and \( f(x) - f(y) \le 0 \). The area of R is equal to
(a) \( 16\pi \)
(b) \( 12\pi \)
(c) \( 8\pi \)
(d) \( 4\pi \)
Answer: (c) \( 8\pi \)

Question. The value of 'a' (\( a > 0 \)) for which the area bounded by the curves \( y = \frac{x}{6} + \frac{1}{x^2} \), y = 0, x = a and x = 2a has the least value, is
(a) 2
(b) \( \sqrt{2} \)
(c) \( 2^{1/3} \)
(d) 1
Answer: (d) 1

Question. Consider the following regions in the plane
\( R_1 = \{(x, y) : 0 \le x \le 1 \text{ and } 0 \le y \le 1\} \) and
\( R_2 = \{(x,y) : x^2 + y^2 \le 4/3\} \)
The area of the region \( R_1 \cap R_2 \) can be expressed as \( \frac{a\sqrt{3} + b\pi}{9} \), where a and b are integers, then
(a) a = 3
(b) a = 1
(c) b = 1
(d) b = 3
Answer: (d) b = 3

Question. The area of the region of the plane bounded by \( (|x|, |y|) \le 1 \) & \( xy \le \frac{1}{2} \) is
(a) less than \( 4 \ln 3 \)
(b) \( \frac{15}{4} \)
(c) \( 2 + 2 \ln 2 \)
(d) \( 3 + \ln 2 \)
Answer: (a) less than \( 4 \ln 3 \) and (d) \( 3 + \ln 2 \)

Question. A point P moves inside a triangle formed by A(0, 0), B(2, \( 2\sqrt{3} \)), C(4, 0) such that min (PA, PB, PC) = 2, then the area bounded by the curve traced by P is
(a) \( 3\sqrt{3} - \frac{3\pi}{2} \)
(b) \( 4\sqrt{3} - 2\pi \)
(c) \( \sqrt{3} - \frac{\pi}{2} \)
(d) \( 2\pi \)
Answer: (b) \( 4\sqrt{3} - 2\pi \)

Question. Area of the region enclosed between the curves \( x = y^2 - 1 \) and \( x = |y| \sqrt{1 - y^2} \) is
(a) 1
(b) \( \frac{4}{3} \)
(c) \( \frac{2}{3} \)
(d) 2
Answer: (d) 2

Question. If the tangent to the curve \( y = 1 - x^2 \) at \( x = \alpha \), where \( 0 < \alpha < 1 \), meets the axes at P and Q. Also \( \alpha \) varies, the minimum value of the area of the triangle OPQ is k times area bounded by the axes and the part of the curve for which \( 0 < x < 1 \), then k is equal to
(a) \( \frac{2}{\sqrt{3}} \)
(b) \( \frac{75}{16} \)
(c) \( \frac{25}{18} \)
(d) \( \frac{2}{3} \)
Answer: (a) \( \frac{2}{\sqrt{3}} \)