Volume of Solids Using Integration Worksheet

Question 1: Find the volume of the solid formed by rotating the curve \( y = x^2 \) about the x-axis, from \( x = 0 \) to \( x = 2 \). \[ V = \pi \int_0^2 (x^2)^2 \, dx \]

Question 2: Find the volume of the solid formed by rotating the curve \( y = 3x \) about the x-axis, from \( x = 1 \) to \( x = 4 \). \[ V = \pi \int_1^4 (3x)^2 \, dx \]

Question 3: Find the volume of the solid formed by rotating the curve \( y = x^3 \) about the y-axis, from \( x = 0 \) to \( x = 1 \). \[ V = 2\pi \int_0^1 x^3 \cdot x \, dx \]

Question 4: Find the volume of the solid formed by rotating the curve \( y = \sqrt{x} \) about the x-axis, from \( x = 1 \) to \( x = 4 \). \[ V = \pi \int_1^4 (\sqrt{x})^2 \, dx \]

Question 5: Find the volume of the solid formed by rotating the curve \( y = 2x - 1 \) about the x-axis, from \( x = 0 \) to \( x = 3 \). \[ V = \pi \int_0^3 (2x - 1)^2 \, dx \]

Question 6: Find the volume of the solid formed by rotating the curve \( y = 4 - x^2 \) about the x-axis, from \( x = -2 \) to \( x = 2 \). \[ V = \pi \int_{-2}^2 (4 - x^2)^2 \, dx \]

Question 7: Find the volume of the solid formed by rotating the curve \( y = x^2 + 1 \) about the y-axis, from \( x = 1 \) to \( x = 3 \). \[ V = 2\pi \int_1^3 (x^2 + 1) \cdot x \, dx \]

Question 8: Find the volume of the solid formed by rotating the curve \( y = 5 - x \) about the x-axis, from \( x = 0 \) to \( x = 5 \). \[ V = \pi \int_0^5 (5 - x)^2 \, dx \]

Question 9: Find the volume of the solid formed by rotating the curve \( y = 1 - x^2 \) about the x-axis, from \( x = 0 \) to \( x = 1 \). \[ V = \pi \int_0^1 (1 - x^2)^2 \, dx \]

Question 10: Find the volume of the solid formed by rotating the curve \( y = e^x \) about the x-axis, from \( x = 0 \) to \( x = 2 \). \[ V = \pi \int_0^2 (e^x)^2 \, dx \]