Evaluate the following integral: ∫ 0 3 4 d x \int_0^3\!4\,dx

in this problem, we are asked to evaluate the following indefinite integral. Now we've learned from the fundamental theorem of calculus part two that we can do these by combining the idea of taking an antiderivative and just evaluating these bounds together. So what we can do is say, alright. So we have this four here that we're integrating from zero to three with respect to x. Now we know that the antiderivative of a constant with respect to x is just gonna be that constant times x. So the antiderivative here is just gonna be four x. Then I need to go ahead and bound this from zero to three. So we've gone ahead and applied this rule. Now what I need to do is plug in the high bound and the low bound and subtract. So we'll start with the high bound. So the high bound is gonna give us four times three. Then we're going to have minus, and I'll plug in the low bound, which is gonna be four times zero. Notice I'm just taking this x here, and they're replacing it with three and with zero. Now also something I wanna mention here is notice I did not add a plus c to this because we don't need to do that when we're dealing with definite integrals. So we have four times three minus four times zero. Four times three is 12, then we have that minus zero, and the whole thing is just gonna come out to 12, meaning 12 is the solution to our problem. So that's how we can find this definite integral. Hope you found this video helpful.

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8. Definite Integrals

Fundamental Theorem of Calculus

Calculus

8. Definite Integrals

Fundamental Theorem of Calculus

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Multiple Choice

Evaluate the following integral:
$integral from 0 to 3 comma 4 d x$

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Verified step by step guidance

Identify the integral to be evaluated: $ \int_0^3 4 \, dx $. This is a definite integral with limits of integration from 0 to 3.

Recognize that the integrand is a constant function, $ f(x) = 4 $. The integral of a constant $ c $ with respect to $ x $ over an interval $[a, b]$ is given by $ c(b-a) $.

Apply the formula for the integral of a constant: $ \int_a^b c \, dx = c(b-a) $. Here, $ c = 4 $, $ a = 0 $, and $ b = 3 $.

Substitute the values into the formula: $ 4(3-0) $.

Simplify the expression to find the value of the integral.