Application of Integrals

What is the integral of x^2 with respect to x?

• (x^3)/3 + C

• x^3 + C

• (x^4)/4 + C

• (x^2)/2 + C

Calculate the definite integral of 3x from 1 to 4.

• 30

• 22.5

• 15

• 10

Find the area under the curve y = x^3 from x = 0 to x = 2.

• 2

• 6

• 4

• 8

Evaluate the integral of sin(x) dx.

• sin(x) + C

• tan(x) + C

• -sin(x) + C

• -cos(x) + C

What is the integral of e^x with respect to x?

• e^x + 1

• e^x - C

• e^x / x

• e^x + C

Determine the volume of the solid formed by rotating y = x^2 around the x-axis from x = 0 to x = 1.

• π/5

• π/6

• π/3

• π/4

Calculate the integral of 1/x from x = 1 to x = 2.

• ln(3)

• 1

• ln(1)

• ln(2)

What is the relationship between the derivative and the integral?

• The derivative is a type of integral.

• The integral measures the rate of change.

• The derivative and the integral are inverse operations.

• The derivative and the integral are unrelated concepts.

Evaluate the integral of cos(x) from 0 to π/2.

• 0

• 2

• 1

• π/2

Find the average value of the function f(x) = x^2 on the interval [1, 3].

• 4.0

• 5.0

• 4.333333333333333

• 3.5