# Application of Integrals

**Exercise 1**

**1.** Find the area of the region bounded by the curve y^{2} = x and the lines x = 1, x = 4 and the x-axis.

**2. **Find the area of the region bounded by y^{2} = 9x, x = 2, x = 4 and the x-axis in the first quadrant.

**3. **Find the area of the region bounded by x^{2} = 4y, y = 2, y = 4 and the y-axis in the first quadrant.

**4.** Find the area of the region bounded by the ellipse

**5. **Find the area of the region bounded by the ellipse

**6. **Find the area of the region in the first quadrant enclosed by x-axis, line x = √3y and the circle x^{2} + y^{2} = 4.

**7. **Find the area of the smaller part of the circle x^{2} + y^{2} = a^{2} cut off by the line

**8.** The area between x = y^{2} and x = 4 is divided into two equal parts by the line x = a, find the value of a.

**9. ** Find the area of the region bounded by the parabola y = x^{2} and y = |x|.

**10.** Find the area bounded by the curve x^{2} = 4y and the line x = 4y – 2.

**11. **Find the area of the region bounded by the curve y^{2} = 4x and the line x = 3.

**12.** Choose the correct answer:

Area lying in the first quadrant and bounded by the circle x^{2} + y^{2} = 4 and the lines x = 0 and x = 2 is

**13.** Choose the correct answer:

Area of the region bounded by the curve y = 4x, y-axis and the line y = 3 is:

**Exercise 2**

**14.** Find the area of the circle 4x^{2} + 4y^{2} = 9 which is interior to the parabola x^{2} = 4y.

**15.** Find the area bounded by the curves (x – 1)^{2} + y^{2} = 1 and x^{2} + y^{2} = 1.

**16.** Find the area of the region bounded by the curves y = x^{2} + 2, y = x, x = 0 and x = 3.

**17.** Using integration, find the area of the region bounded by the triangle whose vertices are (-1, 0), (1, 3) and (3, 2).

**18.** Using integration, find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x = 4.

**19.** Choose the correct answer:

Smaller area enclosed by the circle x + y = 4 and the line x + y = 2 is:

(A) 2(π – 2)

(B) π – 2

(C) 2π – 1

(D) 2(π + 2)

**20.** Choose the correct answer:

Area lying between the curves y^{2} = 4x and y = 2x is:

**Exercise 3**

**21.** Find the area under the given curves and given lines:

(i) y = x^{2}, x = 1, x = 2 and x-axis.

(ii) y = x^{4}, x = 1, x = 2 and x-axis.

**22.** Find the area between the curves y = x and y = x^{2}.

**23.** Find the area of the region lying in the first quadrant and bounded by y = 4x^{4}, x = 0, y = 1 and y = 4.

**24.** Sketch the graph of y = |x + 3| and evaluate

**25.** Find the area bounded by the curve y = sin x between x = 0 and x = 2π.

**26.** Find the area enclosed by the parabola y^{2} = 4ax and the line y = mx.

**27.** Find the area enclosed by the parabola 4y = 3x^{2} and the line 2y = 3x + 12.

**28.** Find the area of the smaller region bounded by the ellipse and the line

**29.** Find the area of the smaller region bounded by the ellipse and the line

**30. ** Find the area of the region enclosed by the parabola x^{2} = y, the line y = x + 2 and x-axis.

**31.** Using the method of integration, find the area enclosed by the curve |x|+|y| = 1.

**32.** Find the area bounded by the curve |(x, y): y ≥ x^{2} and y = |x||.

**33.** Using the method of integration, find the area of the triangle ABC whose vertices are A(2, 0), B(4, 5) and C(6, 3).

**34.** Using the method of integration, find the area of the region bounded by the lines: 2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0.

**35.** Find the area of the region {(x, y): y^{2} ≤ 4x and 4x^{2} + 4y^{2} ≤ 9}.

**36.** Choose the correct answer:

Area bounded by the curve y = x^{3} the x-axis and the ordinate x = -2 and x = 1 is:

**37.** Choose the correct answer:

The area bounded by the curve y = x|x|, x-axis and the ordinates x = -1 and x = 1 is given by:

**38.** Choose the correct answer:

The area of the circle x + y = 16 exterior to the parabola y = 6x.

**39.** Choose the correct answer:

The area bounded by the y-axis, y = cos x and y = sin x when is:

(A) 2(√2 – 1)

(B) √2 – 1

(C) √2 + 1

(D) √2