CBSE Class 12 Mathematics Application of Integrals MCQs Set 05

Question. The area bounded by the curve \( y = x^2 + 4x + 5 \), the axes of coordinates and minimum ordinate is
(a) \( 3\frac{2}{3} \) sq. units
(b) \( 4\frac{2}{3} \) sq. units
(c) \( 5\frac{2}{3} \) sq. units
(d) None of the options
Answer: (b) \( 4\frac{2}{3} \) sq. units

Question. Area of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is
(a) \( 4\pi ab \) sq. units
(b) \( 2\pi ab \) sq. units
(c) \( \pi ab \) sq. units
(d) \( \frac{\pi ab}{2} \) sq. units
Answer: (c) \( \pi ab \) sq. units

Question. The area bounded by the curve \( 2x^2 + y^2 = 2 \) is
(a) \( \pi \) sq. units
(b) \( \sqrt{2}\pi \) sq. units
(c) \( \frac{\pi}{2} \) sq. units
(d) \( 2\pi \) sq. units
Answer: (b) \( \sqrt{2}\pi \) sq. units

Question. Area enclosed by the circle \( x^2 + y^2 = a^2 \) is equal to
(a) \( 2\pi a^2 \) sq. units
(b) \( \pi a^2 \) sq. units
(c) \( 2\pi a \) sq. units
(d) \( \pi a \) sq. units
Answer: (b) \( \pi a^2 \) sq. units

Question. Area bounded by the ellipse \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) is
(a) \( 6\pi \) sq. units
(b) \( 3\pi \) sq. units
(c) \( 12\pi \) sq. units
(d) None of the options
Answer: (a) \( 6\pi \) sq. units

Question. The area enclosed between the curve \( x^2 + y^2 = 16 \) and the coordinate axes in the first quadrant is
(a) \( 4\pi \) sq. units
(b) \( 3\pi \) sq. units
(c) \( 2\pi \) sq. units
(d) \( \pi \) sq. units
Answer: (a) \( 4\pi \) sq. units

Question. The area enclosed by the curve \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \) is
(a) \( 10\pi \) sq. units
(b) \( 15\pi \) sq. units
(c) \( 5\pi \) sq. units
(d) \( 4\pi \) sq. units
Answer: (b) \( 15\pi \) sq. units

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

Question. The area of the region bounded by the parabola \( y = x^2 + 1 \) and the straight line \( x + y = 3 \) is given by
(a) \( \frac{45}{7} \) sq. units
(b) \( \frac{25}{4} \) sq. units
(c) \( \frac{5}{18} \) sq. units
(d) \( \frac{9}{2} \) sq. units
Answer: (d) \( \frac{9}{2} \) sq. units

Question. The area enclosed between the curve \( y^2 = 4x \) and the line \( y = x \) is
(a) \( \frac{8}{3} \) sq. units
(b) \( \frac{4}{3} \) sq. units
(c) \( \frac{2}{3} \) sq. units
(d) \( \frac{1}{2} \) sq. units
Answer: (a) \( \frac{8}{3} \) sq. units

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

Question. Area of the region bounded by the curve \( y = x^2 \) and the line \( y = 4 \) is
(a) \( \frac{11}{3} \) sq. units
(b) \( \frac{32}{3} \) sq. units
(c) \( \frac{43}{3} \) sq. units
(d) \( \frac{47}{3} \) sq. units
Answer: (b) \( \frac{32}{3} \) sq. units

Question. Area lying between the parabola \( y^2 = 4x \) and its latus rectum is
(a) \( \frac{1}{3} \) sq. units
(b) \( \frac{2}{3} \) sq. units
(c) \( \frac{5}{3} \) sq. units
(d) \( \frac{8}{3} \) sq. units
Answer: (d) \( \frac{8}{3} \) sq. units

Question. The area bounded by the curve \( y^2 = x \), line \( y = 4 \) and y-axis is
(a) \( \frac{16}{3} \) sq. units
(b) \( \frac{64}{3} \) sq. units
(c) \( 7\sqrt{2} \) sq. units
(d) None of the options
Answer: (b) \( \frac{64}{3} \) sq. units

Question. The area bounded by the curve \( x = 3y^2 - 9 \) and the line \( x = 0 \), \( y = 0 \) and \( y = 1 \) is
(a) \( 8 \) sq. units
(b) \( 8/3 \) sq. units
(c) \( 3/8 \) sq. unit
(d) \( 3 \) sq. units
Answer: (a) \( 8 \) sq. units

Question. Find the area above \( x \)-axis, bounded by the curves \( y = 2^{kx} \), \( x = 0 \) and \( x = 2 \).
(a) \( \frac{4^k - 1}{k \log_e 2} \)
(b) \( \frac{2^k - 1}{2\log_e 2} \)
(c) \( \frac{3 - k}{k \log_e 2} \)
(d) \( \frac{-1 + 3^k}{2\log_e 2} \)
Answer: (a) \( \frac{4^k - 1}{k \log_e 2} \)

Question. Find the area enclosed by the parabola \( y^2 = x \) and the line \( y + x = 2 \) and the \( x \)-axis.
(a) \( \frac{5}{6} \) sq. units
(b) \( \frac{7}{6} \) sq. units
(c) \( \frac{6}{7} \) sq. units
(d) \( \frac{4}{7} \) sq. units
Answer: (b) \( \frac{7}{6} \) sq. units

Question. The area bounded by the curve \( x^2 + y^2 = 1 \) in first quadrant is
(a) \( \frac{\pi}{4} \) sq. units
(b) \( \frac{\pi}{2} \) sq. units
(c) \( \frac{\pi}{3} \) sq. units
(d) \( \frac{\pi}{6} \) sq. units
Answer: (a) \( \frac{\pi}{4} \) sq. units

Question. Area bounded by the curve \( y = \cos x \) between \( x = 0 \) and \( x = \frac{3\pi}{2} \) is
(a) \( 1 \) sq. unit
(b) \( 2 \) sq. units
(c) \( 3 \) sq. units
(d) \( 4 \) sq. units
Answer: (c) \( 3 \) sq. units

Question. Area of the region bounded by the curve \( y = \tan x \), line \( x = \frac{\pi}{4} \) and the \( x \)-axis is
(a) \( \log 2 \) sq. units
(b) \( \frac{1}{2}\log 2 \) sq. units
(c) \( \frac{1}{3}\log 2 \) sq. units
(d) \( 5 \log 2 \) sq. units
Answer: (b) \( \frac{1}{2}\log 2 \) sq. units

Question. The area bounded by the curve \( y = \sec^2 x \), \( y = 0 \) and \( |x| = \frac{\pi}{3} \) is
(a) \( \sqrt{3} \) sq. units
(b) \( \sqrt{2} \) sq. units
(c) \( 2\sqrt{3} \) sq. units
(d) None of the options
Answer: (c) \( 2\sqrt{3} \) sq. units

Question. The area bounded by the curve \( x^2 = 4y + 4 \) and line \( 3x + 4y = 0 \) is
(a) \( \frac{25}{4} \) sq. units
(b) \( \frac{125}{8} \) sq. units
(c) \( \frac{125}{16} \) sq. units
(d) \( \frac{125}{24} \) sq. units
Answer: (d) \( \frac{125}{24} \) sq. units

Question. The area bounded by the \( x \)-axis, the curve \( y = f(x) \) and the lines \( x = 1 \), \( x = b \) is equal to \( \sqrt{b^2 + 1} - \sqrt{2} \) for all \( b > 1 \), then \( f(x) \) is
(a) \( \sqrt{x - 1} \)
(b) \( \sqrt{x + 1} \)
(c) \( \sqrt{x^2 + 1} \)
(d) \( x / \sqrt{x^2 + 1} \)
Answer: (d) \( x / \sqrt{x^2 + 1} \)

Question. The area (in sq. units) enclosed between the graph of \( y = x^3 \) and the lines \( x = 0 \), \( y = 1 \), \( y = 8 \) is
(a) \( \frac{45}{4} \)
(b) \( 14 \)
(c) \( 7 \)
(d) None of the options
Answer: (a) \( \frac{45}{4} \)

Question. The area of the region bounded by the curve \( y = \sqrt{16 - x^2} \) and \( x \)-axis is
(a) \( 8\pi \) sq. units
(b) \( 20\pi \) sq. units
(c) \( 16\pi \) sq. units
(d) \( 256\pi \) sq. units
Answer: (a) \( 8\pi \) sq. units

Question. Area of the region bounded by the curve \( y = \cos x \) between \( x = 0 \) and \( x = \pi \) is
(a) \( 2 \) sq. units
(b) \( 4 \) sq. units
(c) \( 3 \) sq. units
(d) \( 1 \) sq. unit
Answer: (a) \( 2 \) sq. units

Question. The area of the region bounded by parabola \( y^2 = x \) and the straight line \( 2y = x \) is
(a) \( \frac{4}{3} \) sq. units
(b) \( 1 \) sq. unit
(c) \( \frac{2}{3} \) sq. unit
(d) \( \frac{1}{3} \) sq. unit
Answer: (a) \( \frac{4}{3} \) sq. units

Question. The area of the region bounded by the curve \( y = \sin x \) between the ordinates \( x = 0 \), \( x = \frac{\pi}{2} \) and the \( x \)-axis is
(a) \( 2 \) sq. units
(b) \( 4 \) sq. units
(c) \( 3 \) sq. units
(d) \( 1 \) sq. unit
Answer: (d) \( 1 \) sq. unit

Question. The area of the region bounded by the ellipse \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \) is
(a) \( 20\pi \) sq. units
(b) \( 20\pi^2 \) sq. units
(c) \( 16\pi^2 \) sq. units
(d) \( 25\pi \) sq. units
Answer: (a) \( 20\pi \) sq. units

Question. The area of the region bounded by the circle \( x^2 + y^2 = 1 \) is
(a) \( 2\pi \) sq. units
(b) \( \pi \) sq. units
(c) \( 3\pi \) sq. units
(d) \( 4\pi \) sq. units
Answer: (b) \( \pi \) sq. units

Assertion & Reasoning Based MCQs

Question. Assertion : The area of the region bounded by the curve \( y^2 = 4x \) and the line \( x = 3 \) is \( 8\sqrt{3} \) sq. units.
Reason : The area of the region bounded by the curve \( x^2 = 4y \) and the line \( x = 4y - 2 \) is \( \frac{9}{8} \) sq. units.
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Question. Assertion : The area of the smaller region bounded by the ellipse \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \) and the line \( \frac{x}{3} + \frac{y}{2} = 1 \) is \( \frac{3}{2}(\pi - 2) \) sq. units.
Reason : Formula to calculate the area of the smaller region bounded by the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and the line \( \frac{x}{a} + \frac{y}{b} = 1 \) is \( \frac{ab}{4} (\pi - 2) \) sq. units.
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Question. Assertion : The area bounded by the parabola \( y^2 = 4ax \) and the line \( x = a \) and \( x = 4a \) is \( \frac{56a^2}{3} \) sq. units.
Reason : The area bounded by the curves \( y = 3x \) and \( y = x^2 \) is 9.5 sq. units.
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (c) Assertion is correct statement but Reason is wrong statement.

Question. Assertion : The area bounded by the curves \( y^2 = 4a^2(x - 1) \) and lines \( x = 1 \) and \( y = 4a \) is \( \frac{8a}{3} \) sq. units.
Reason : The area enclosed between the parabola \( y = x^2 - x + 2 \) and the line \( y = x + 2 \) is \( \frac{4}{3} \) sq. units.
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (d) Assertion is wrong statement but Reason is correct statement.

Question. Assertion : The area bounded by the curve \( y = 2\cos x \) and the \( x \)-axis from \( x = 0 \) to \( x = 2\pi \) is 8 sq. units.
Reason : The area bounded by the curve \( y = \sin x \) between \( x = \pi \) and \( x = 2\pi \) is 4 sq. units.
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (c) Assertion is correct statement but Reason is wrong statement.