Question

If `int_0^a 3x^2 dx = 8` then a = ______.

Options

• 0

• 2

• 8

• `4/3`

Answer 1

If `int_0^a 3x^2 dx = 8` then a = 2.

Explanation:

1. Find the Antiderivative

To solve the integral `int_0^a 3x^2 dx`, we first determine the antiderivative of the function f(x) = 3x². 

Using the power rule for integration:

`int x^n dx = x^(n + 1)/(n + 1)`

The integral of 3x² is:

`int 3x^2 dx = 3 * x^3/3`

= x³

2. Apply the Fundamental Theorem of Calculus

Next, we evaluate the definite integral by applying the upper limit  (a) and lower limit (0) to the antiderivative:

`[x^3]_0^a = a^3 - 0^3`

= a³

3. Solve for a

According to the problem, the value of the integral is 8. We set up the following equation:

a³ = 8

Taking the cube root of both sides gives:

`a = root(3)(8)`

a = 2