Math / Calculus

QuestionEvaluate the integral. Do not use your calculator to find/simplify your answer (4 points) $\int_{0}^{4} d x$

Studdy Solution

STEP 1

1. The integral $\int_{0}^{4} dx$ represents the area under the curve of the function $f(x) = 1$ from $x = 0$ to $x = 4$ .
2. The function $f(x) = 1$ is a constant function.

STEP 2

1. Identify the function being integrated.
2. Determine the antiderivative of the function.
3. Evaluate the definite integral using the limits of integration.

STEP 3

Identify the function being integrated. In this case, the integrand is simply $1$ , which means we are integrating the constant function:
$f(x) = 1$

STEP 4

Determine the antiderivative of the function. The antiderivative of a constant $c$ is $cx$ . Thus, the antiderivative of $1$ is:
$F(x) = x$

STEP 5

Evaluate the definite integral using the limits of integration from $0$ to $4$ . We apply the Fundamental Theorem of Calculus:
$\int_{0}^{4} dx = F(4) - F(0)$
Substitute the antiderivative into the expression:
$F(4) = 4$ $F(0) = 0$
Calculate the difference:
$\int_{0}^{4} dx = 4 - 0 = 4$
The value of the integral is:
$\boxed{4}$

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