CBSE Class 12 Mathematics Application of Integrals VBQs Set A

Multiple Choice Questions

Question. The area enclosed by the circle \(x^2 + y^2 = 2\) is equal to
(a) \(4\pi\) sq units
(b) \(2\sqrt{2}\pi\) sq units
(c) \(4\pi^2\) sq units
(d) \(2\pi\) sq units
Answer: (d)

Question. The area enclosed by the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is equal to 
(a) \(\pi^2 ab\) sq units
(b) \(\pi ab\) sq units
(c) \(\pi a^2 b\) sq units
(d) \(\pi ab^2\) sq units
Answer: (b)

Question. The area of the region bounded by the y-axis, \(y = \cos x\) and \(y = \sin x\), \(0 \leq x \leq \frac{\pi}{2}\) is 
(a) \(\sqrt{2}\) sq units
(b) \((\sqrt{2} + 1)\) sq units
(c) \((\sqrt{2} - 1)\) sq units
(d) \((2\sqrt{2} - 1)\) sq units
Answer: (c)

Question. The area of the region bounded by the curve \(x^2 = 4y\) and the straight line \(x = 4y - 2\) is 
(a) \(\frac{3}{8}\) sq unit
(b) \(\frac{5}{8}\) sq unit
(c) \(\frac{7}{8}\) sq unit
(d) \(\frac{9}{8}\) sq units
Answer: (d)

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)

Question. Area of the region in the first quadrant enclosed by the x-axis, the line \(y = x\) and the circle \(x^2 + y^2 = 32\) is 
(a) \(16\pi\) sq units
(b) \(4\pi\) sq units
(c) \(32\pi\) sq units
(d) \(24\pi\) sq units
Answer: (b)

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)

Question. The area of the region bounded by the curve \(x = 2y + 3\) and the lines \(y = 1\) and \(y = -1\) is 
(a) \(4\) sq units
(b) \(\frac{3}{2}\) sq units
(c) \(6\) sq units
(d) \(8\) sq units
Answer: (c)

Question. The area of the region bounded by the curve \(y = x^2\) and the line \(y = 16\) is
(a) \(\frac{37}{3}\) sq units
(b) \(\frac{256}{3}\) sq units
(c) \(\frac{64}{3}\) sq units
(d) \(\frac{128}{3}\) sq units
Answer: (b)

Question. The area of the region bounded by the curve \(y^2 = 9x, y = 3x\) is
(a) \(1\) sq unit
(b) \(\frac{1}{2}\) sq unit
(c) \(4\) sq units
(d) \(14\) sq units
Answer: (b)

Question. The area of the curve \(y = \sin x\) between 0 and \(\pi\) is
(a) \(2\) sq units
(b) \(4\) sq units
(c) \(12\) sq units
(d) \(14\) sq units
Answer: (a)

Question. The area of the region bounded by the curve \(ay^2 = x^3\), the y-axis and the lines \(y = a\) and \(y = 2a\) is
(a) \(3\) sq units
(b) \(\frac{3}{5} a^{2} [2 \times 2^{2/3} - 1]\) sq units
(c) \(\frac{3}{5} a [2^{3/2} - 1]\) sq units
(d) \(1\) sq unit
Answer: (b)

Question. The area enclosed by the curve \(x = 3 \cos t, y = 2 \sin t\) is
(a) \(4\pi\) sq units
(b) \(6\pi\) sq units
(c) \(14\pi\) sq units
(d) \(7\pi\) sq units
Answer: (b)

Question. The area of the region bounded by the curves \(x = at^2\) and \(y = 2at\) between the ordinate corresponding to \(t = 1\) and \(t = 2\) is
(a) \(\frac{56}{3} a^2\) sq units
(b) \(\frac{40}{3} a^2\) sq units
(c) \(5\pi\) sq units
(d) None of these
Answer: (a)

Question. The area of a minor segment of the circle \(x^2 + y^2 = a^2\) cut off by the line \(x = \frac{a}{2}\) is
(a) \(\frac{a^2}{12} (4\pi - 3\sqrt{3})\) sq units
(b) \(\frac{a^2}{4} (4\pi - 3)\) sq units
(c) \(\frac{a^2}{12} (3\pi - 4)\) sq units
(d) None of these
Answer: (a)

Question. The area of the region bounded by the curve \(y = x^3\) and \(y = x + 6\) and \(x = 0\) is
(a) \(7\) sq units
(b) \(6\) sq units
(c) \(10\) sq units
(d) \(14\) sq units
Answer: (c)

Question. The area under the curve \(y = 2\sqrt{x}\) included between the lines \(x = 0\) and \(x = 1\) is
(a) \(4\) sq units
(b) \(3\) sq units
(c) \(\frac{4}{3}\) sq units
(d) None of these
Answer: (c)

Question. The area under the curve \(y = \sqrt{a^2 - x^2}\) included between the lines \(x = 0\) and \(x = a\) is
(a) \(\frac{\pi a^2}{4}\) sq units
(b) \(\frac{a^2}{4}\) sq units
(c) \(\pi a^2\) sq units
(d) \(4\pi\) sq units
Answer: (a)

Question. The area of the region bounded by the triangle whose vertices are \((-1, 1), (0, 5)\) and \((3, 2)\) is
(a) \(\frac{15}{2}\) sq units
(b) \(15\) sq units
(c) \(4\) sq units
(d) \(10\) sq units
Answer: (a)

Question. The area of the region bounded by the line \(y - 1 = x\), the x-axis and the ordinates \(x = -2\) and \(x = 3\) is
(a) \(\frac{4}{3}\) sq units
(b) \(\frac{7}{2}\) sq units
(c) \(\frac{17}{2}\) sq units
(d) \(\frac{16}{3}\) sq units
Answer: (c)