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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0. Functions7h 54m

Introduction to Functions
16m

Piecewise Functions
10m

Properties of Functions
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Common Functions
1h 8m

Transformations
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Combining Functions
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Exponent rules
32m

Exponential Functions
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Logarithmic Functions
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Properties of Logarithms
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Exponential & Logarithmic Equations
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Introduction to Trigonometric Functions
38m

Graphs of Trigonometric Functions
44m

Trigonometric Identities
47m

Inverse Trigonometric Functions
48m

1. Limits and Continuity2h 2m

Introduction to Limits
41m

Finding Limits Algebraically
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Continuity
40m

2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
30m

Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
26m

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Basic Rules of Differentiation
39m

Higher Order Derivatives
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40m

The Chain Rule
49m

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59m

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5. Graphical Applications of Derivatives6h 2m

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Finding Global Extrema
50m

The First Derivative Test
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Concavity
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The Second Derivative Test
31m

Curve Sketching
44m

Applied Optimization
1h 25m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m

Derivatives of Exponential & Logarithmic Functions
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Logarithmic Differentiation
20m

Derivatives of Inverse Trigonometric Functions
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7. Antiderivatives & Indefinite Integrals1h 26m

Antiderivatives
17m

Indefinite Integrals
25m

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15m

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8. Definite Integrals4h 44m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 21m

Introduction to Definite Integrals
22m

Fundamental Theorem of Calculus
37m

Average Value of a Function
21m

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11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m

Integrals of Exponential Functions
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58m

Integrals Involving Inverse Trigonometric Functions
55m

12. Techniques of Integration7h 41m

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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.