CBSE Class 12 Mathematics Application of Integrals Important Questions Set 01

Objective Type Questions:

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

Question. The area bounded by the curve \(y = |\sin x|\), x-axis and ordinates \(x = \pi\) and \(x = 10\pi\) is equal to
(a) 8 sq. units
(b) 10 sq. units
(c) 18 sq. units
(d) 20 sq. units
Answer: (c)

Question. The area of the region bounded by the 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)

Question. The area of the circle \(x^2 + y^2 = 16\) exterior to the parabola \(y^2 = 6x\) is
(a) \(\frac{4}{3}(4\pi - \sqrt{3})\) sq. units
(b) \(\frac{4}{3}(4\pi + \sqrt{3})\) sq. units
(c) \(\frac{4}{3}(8\pi - \sqrt{3})\) sq. units
(d) \(\frac{4}{3}(8\pi + \sqrt{3})\) sq. units
Answer: (c)

Question. Area lying in the first quadrant and bounded by the circle \(x^2 + y^2 = 4\) and the line \(x = 0\) and \(x = 2\) is
(a) \(\pi\) sq. units
(b) \(\frac{\pi}{2}\) sq. units
(c) \(\frac{\pi}{3}\) sq. units
(d) \(\frac{\pi}{4}\) sq. units
Answer: (a)

Fill in the blanks.

Question. The area of the region bounded by the curve \(y = x^2 + x\), x-axis and the line \(x = 2\) and \(x = 5\) is equal to _____________ .
Answer: \(\frac{297}{6}\) sq. units

Question. The area bounded by the curve \(y = e^x\), x-axis and ordinates \(x = 0\) and \(x = 2\) is _____________ .
Answer: \((e^2 - 1)\) sq units

Question. The area bounded by the curves \(y = |x|\), and \(x = - 1\) and \(x = 1\) is _____________ .
Answer: 1 sq. unit

Very Short Answer Questions:

Question. Find the area bounded by the curve \(y = x^2\), \(x = 2, x = 3\) and x-axis.
Answer: \(\frac{19}{3}\) sq. units

Question. Calculate the area under the curve \(y = 2\sqrt{x}\) included between the lines \(x = 0\) and \(x = 1\).
Answer: \(\frac{4}{3}\) sq. units

Question. Find the area under the curve \(y = \sqrt{x - 1}\) between the lines \(x = 1\) and \(x = 5\).
Answer: \(\frac{16}{3}\) sq. units

Long Answer Questions:

Question. Find the area bounded by the lines \(y = 4x + 5, y = 5 - x\) and \(4y = x + 5\). 
Answer: \(\frac{15}{2}\) sq. units

Question. Find the area bounded by the curve \(x^2 = 4y\) and the straight line \(x = 4y - 2\).
Answer: \(\frac{9}{8}\) sq. units

Question. Using integration, find the area of the region \(\{(x, y)\} : 9x^2 + y^2 \le 36\) and \(3x + y \ge 6\).
Answer: \(3(\pi - 2)\) sq. units

Question. Find the area of the region \(\{(x, y) : x^2 \le y \le x\}\).
Answer: \(\frac{1}{6}\) sq. unit

Question. Find the area of the region bounded by the curve \(y = \frac{3}{4}x^2\) and the line \(3x - 2y + 12 = 0\).
Answer: 27 sq. units

Question. Using integration, find the area of the triangle ABC, where A is (2, 3), B is (4, 7) and C is (6, 2).
Answer: 9 sq. units

Question. Make a rough sketch of the region given below and find its area, using integration: \(\{(x, y) : 0 \le y \le x^2 + 3; 0 \le y \le 2x + 3, 0 < x \le 3\}\)
Answer: \(\frac{50}{3}\) sq. units

Question. Using integration, find the area of the triangle ABC, whose vertices have coordinates A (2, 0), B (4, 5) and C (6, 3).
Answer: 7 sq. units

Question. Find the area of the smaller region bounded by the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) and the line \(\frac{x}{3} + \frac{y}{2} = 1\). 
Answer: \((\frac{3\pi}{2} - 3)\) sq. units

Question. Using integration, find the area of the triangle formed by negative x-axis and tangent and normal to the circle \(x^2 + y^2 = 9\) at \((-1, 2\sqrt{2})\). 
Answer: \(9\sqrt{2}\)

Question. If the area bounded by the parabola \(y^2 = 16ax\) and the line \(y = 4mx\) is \(\frac{a^2}{12}\) sq units, then using integration, find the value of m. 
Answer: \(m = 2\sqrt{2}\)

Question. Using integration, find the area bounded by the curves \(y = |x - 1|\) and \(y = 3 - |x|\).
Answer: 4 sq. units

Question. Using the method of integration, find the area of the triangular region whose vertices are (2, -2), (4, 3) and (1, 2).
Answer: \(\frac{13}{2}\) sq. units

Question. Using integration, find the area of the region bounded by the curves \(y = \sqrt{4 - x^2}\), \(x^2 + y^2 - 4x = 0\) and the x-axis.
Answer: \((\frac{4\pi}{3} - \sqrt{3})\) sq. units

Question. Find the area of the triangle whose vertices are \((-1, 1), (0, 5)\) and \((3, 2)\), using integration. 
Answer: \(\frac{15}{2}\) sq. units

Question. Using integration, find the area of the triangle whose vertices are (2, 3), (3, 5) and (4, 4).
Answer: \(\frac{3}{2}\) sq. units

Question. Using integration, find the area of the following region: \(\{(x, y) : x^2 + y^2 \le 16a^2 \text{ and } y^2 \le 6ax\}\) 
Answer: \(\frac{4a^2}{3}(4\pi + \sqrt{3})\)

Question. Using integration find the area of the region bounded between the two circles \(x^2 + y^2 = 9\) and \((x - 3)^2 + y^2 = 9\). 
Answer: \(2\left(3\pi - \frac{9\sqrt{3}}{4}\right)\) sq. units

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

Question. Using integration, the area of the region bounded by the line \(2y = 5x + 7\), x-axis and the lines \(x = 2\) and \(x = 8\) is
(a) 90 sq units
(b) 96 sq units
(c) 40 sq units
(d) 10 sq units
Answer: (b)

Question. The area of the parabola \(y^2 = 4ax\) bounded by its latus rectum is
(a) \(\frac{4a^2}{3}\) sq units
(b) \(\frac{8a^2}{3}\) sq units
(c) \(\frac{9a^2}{4}\) sq units
(d) \(\frac{8a^2}{5}\) sq units
Answer: (b)

Fill in the blanks.

Question. The area bounded by \(x = 4 - y^2\) and y-axis is _____________ .
Answer: \(\frac{32}{3}\) sq. units

Question. The area between x-axis and the curve \(y = \cos x\) when \(0 \le x \le 2\pi\), is _____________ .
Answer: 4 sq. units

Solve the following questions.

Question. If the area above x-axis, bounded by curves \(y = 2^{kx}\), \(x = 0\) and \(x = 2\) is \(\frac{3}{\log e^2}\), then find the value of k.
Answer: \(k = 1\)

Question. Find the area common to parabola \(y = 2x^2\) and \(y = x^2 + 4\).
Answer: 4 sq. units

Question. Find the area of the region \(\{(x, y) : x^2 + y^2 \le 1 \le x + y\}\).
Answer: \(\frac{1}{2}(\pi - 1)\) sq. units

Question. Find the area bounded by parabola \(y^2 = x\) and straight line \(2y = x\).
Answer: \(\frac{4}{3}\) sq. units

Question. Find the area of the smaller part of the circle \(x^2 + y^2 = a^2\) cut off by the line \(x = \frac{a}{\sqrt{2}}\).
Answer: \(\frac{a^2}{4}(\pi - 2)\) sq. units

Question. Find the area lying above x-axis and included between the circle \(x^2 + y^2 = 8x\) and inside of the parabola \(y^2 = 4x\).
Answer: \(\frac{4}{3}(8 + 3\pi)\) sq. units

Question. Find the area enclosed by the parabola \(y^2 = x\) and line \(y + x = 2\). 
Answer: \(\frac{9}{2}\) sq. units

Question. Using integration, find the area of the triangle ABC with vertices as A(-1, 0), B(1, 3) and C(3, 2). 
Answer: 4 sq. units

Question. Using integration, find the area of the triangle ABC, where A is (2, 3), B is (4, 7) and C is (6, 2). 
Answer: 9 sq. units