Math / Calculus

Question(1) $\int^{3} x^{2} d x$

Studdy Solution

STEP 1

1. We are given an indefinite integral $\int x^2 \, dx$ .
2. The integral is evaluated from a lower limit of $0$ to an upper limit of $3$ .

STEP 2

1. Find the antiderivative of the function $x^2$ .
2. Evaluate the definite integral using the Fundamental Theorem of Calculus.

STEP 3

Find the antiderivative of $x^2$ .
The antiderivative of $x^2$ is:
$F(x) = \frac{x^3}{3} + C$
where $C$ is the constant of integration.

STEP 4

Apply the Fundamental Theorem of Calculus to evaluate the definite integral from $0$ to $3$ .
$\int_{0}^{3} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{3}$

STEP 5

Evaluate the antiderivative at the upper limit $x = 3$ .
$F(3) = \frac{3^3}{3} = \frac{27}{3} = 9$

STEP 6

Evaluate the antiderivative at the lower limit $x = 0$ .
$F(0) = \frac{0^3}{3} = 0$

STEP 7

Subtract the value of the antiderivative at the lower limit from the value at the upper limit.
$\int_{0}^{3} x^2 \, dx = F(3) - F(0) = 9 - 0 = 9$
The value of the definite integral is:
$\boxed{9}$

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