Application of Integrals quiz

The area of the region bounded by the circle x^2 + y^2 = 4 is:

The area bounded by y = x^2, the x-axis and x = 3 is:

The area of the ellipse x^2/9 + y^2/4 = 1 is:

The area bounded by y = sin x and the x-axis between x = 0 and x = pi is:

The area bounded by y = 2x, the x-axis and x = 3 is:

The area bounded by y = cos x and the x-axis from x = 0 to x = 2 pi is:

The area enclosed by the circle x^2 + y^2 = 9 in the first quadrant is:

The area bounded by y = x^3, the x-axis and the lines x = -1 and x = 1 is:

The area bounded by y = e^x, the x-axis and the lines x = 0 and x = 1 is:

The area bounded by x = y^2, the y-axis and the lines y = 0 and y = 3 is:

Find the area bounded by y = 4 - x, the x-axis and the y-axis. (2 marks)

Find the area bounded by y = x^2, the x-axis and the lines x = 0 and x = 2. (2 marks)

Find the area bounded by y = 1/x, the x-axis and the lines x = 1 and x = e. (2 marks)

Find the area under y = sqrt(x) between x = 0 and x = 1. (2 marks)

Find the area bounded by y = sec^2 x, the x-axis and x = 0 to x = pi/4. (2 marks)

Find the area bounded by x = 2y, the y-axis and the lines y = 1 and y = 3. (2 marks)

Find the area bounded by y = abs(x), the x-axis and the lines x = -2 and x = 2. (2 marks)

Find the area of the region bounded by the ellipse x^2/25 + y^2/4 = 1 in the first quadrant. (2 marks)

Why is the integral of sin x from 0 to 2 pi zero although the area is not zero? (2 marks)

Find the area bounded by y = x + 1, the x-axis and the lines x = 0 and x = 2. (2 marks)

Find the area of the region bounded by y = x^2, the x-axis and the lines x = 1 and x = 2. (3 marks)

Find the area under the curve y = sqrt(x) from x = 0 to x = 4. (3 marks)

Using integration, find the area of the region in the first quadrant enclosed by the circle x^2 + y^2 = 25. (3 marks)

Find the area bounded by y = cos x and the x-axis between x = 0 and x = pi/2. Hence find the total area between x = 0 and x = pi. (3 marks)

Find the area bounded by y = 2x + 1, the x-axis and the lines x = 0 and x = 3. (3 marks)

Read the passage and answer the questions. A park in the shape of an ellipse has the boundary x^2/25 + y^2/16 = 1, where x and y are in metres. The municipality wants to plant grass on the whole park. (i) Write the integral for the area in the first quadrant. (ii) Find the total area of the park. (iii) Find the cost of grassing at Rs 50 per m^2 (take pi = 3.14). (5 marks)

Read the passage and answer the questions. The entrance gate of a fort is a parabolic arch given by y = 4 - x^2 above the ground (y = 0), where x and y are in metres. (i) Find where the arch meets the ground. (ii) Write the integral for the area of the gate opening. (iii) Find this area. (5 marks)

Read the passage and answer the questions. A circular pond has the boundary x^2 + y^2 = 36 (in metres). A gardener wants to find its area by integration. (i) Express y in terms of x for the upper semicircle. (ii) Write the integral for the area of the quarter in the first quadrant. (iii) Find the area of the whole pond. (5 marks)

Read the passage and answer the questions. A logo is designed using the curve y = sin x from x = 0 to x = 2 pi. A student computes the integral of sin x dx from 0 to 2 pi and gets 0. (i) Explain the mistake. (ii) Find the correct area. (iii) Sketch the region. (5 marks)

Read the passage and answer the questions. A water channel has a cross-section bounded by y = x^2/4 and the line y = 4 (in metres). (i) Find the points where the line meets the curve. (ii) Find the area of the cross-section using horizontal strips. (iii) Find the volume of water in a 10 m length of the channel when full. (5 marks)

Find the area of the region bounded by the ellipse x^2/a^2 + y^2/b^2 = 1 with a sketch. (6 marks)

Find the area enclosed by the circle x^2 + y^2 = a^2 using integration, with a neat sketch. (6 marks)

Find the area of the region bounded by the curve y^2 = 4x and the line x = 3, with a sketch. (6 marks)

Find the area of the region bounded by the curve y = x^2 and the line y = 4, with a sketch showing horizontal strips. (6 marks)

Sketch the graph of y = abs(x + 3) and evaluate the integral of abs(x + 3) dx from -6 to 0. Interpret the result as an area. (6 marks)