Find the area under the curve in the function below from x = − 4 x=-4 x = − 4 to x = 8 x=8 .

see how we can solve this problem. So here we're asked to find the area under the curve in the function below, and we're asked to find this from x equals negative 4 to x equals positive 8. Now to do this, what I'm going to do is take a look at how our graph is going to behave. So I'm gonna start here at x equals negative 4. That's the start of the interval. We're gonna finish over here at x equals positive 8. So the area that we're calculating is going to be this whole area right in here, plus the area that we have back here. And I noticed that this area goes beneath the x axis right here. So what that means is that area is actually gonna be negative. We're gonna call this area 1, then we're gonna call this area 2. And the total area is going to be the sum of area 1 and area 2. Now I can see here that clearly we have triangles that form, and we want to use these familiar shapes when solving these types of problems. So because I see that we have 2 triangles here, what I'm going to do is say that our areas that we're going to be dealing with are going to be 1 half base times height. So that's the area of a triangle. Now starting with area 1, we'll have 1 half the base here, which I can see goes from 0 to 4. So that's going to be 1 half of 4. And then our height goes down to negative 4. Right? It's gonna be right about there. So our height is going to be negative 4. That's the area of triangle 1. And we're gonna have plus, and then we're gonna have the area of triangle 2, which is 1 half the base, which goes from 0 all the way out to 8 times the height, which goes all the way up to 8, because we can see that that's there on the y axis. So So this is the equation we need. Now one half of 4 is 2, and then one half of 8 is 4. Now 2 times negative 4 is negative 8, and 4 times 8 is positive 32. Now negative 8 +32, well, that's going to give you 24. So 24 is the total area that we're dealing with, and that is the solution to this problem. So hope you found this video helpful. Let's move on.

Table of contents

Skip topic navigation

0. Fundamental Concepts of Algebra3h 32m
1. Equations and Inequalities3h 27m
2. Graphs1h 43m
3. Functions & Graphs2h 17m
4. Polynomial Functions1h 54m
5. Rational Functions1h 23m
6. Exponential and Logarithmic Functions2h 28m
7. Measuring Angles40m
8. Trigonometric Functions on Right Triangles2h 5m
9. Unit Circle1h 19m
10. Graphing Trigonometric Functions1h 19m
11. Inverse Trigonometric Functions and Basic Trig Equations1h 41m
12. Trigonometric Identities 2h 34m
13. Non-Right Triangles1h 38m
14. Vectors2h 25m
15. Polar Equations2h 5m
16. Parametric Equations1h 6m
17. Graphing Complex Numbers1h 7m
18. Systems of Equations and Matrices3h 6m
19. Conic Sections2h 36m
20. Sequences, Series & Induction1h 15m
21. Combinatorics and Probability1h 45m
22. Limits & Continuity1h 49m
23. Intro to Derivatives & Area Under the Curve2h 9m

23. Intro to Derivatives & Area Under the Curve

Area Under a Curve

Precalculus

23. Intro to Derivatives & Area Under the Curve

Area Under a Curve

Previous problemNext problem

Struggling with Precalculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

Find the area under the curve in the function below from $x=-4$ to $x equals 8$ .

Previous problemNext problem

Verified step by step guidance

Identify the function represented by the red line in the graph. It appears to be a linear function, which can be expressed in the form y = mx + b, where m is the slope and b is the y-intercept.

Determine the slope (m) of the line by selecting two points on the line. For example, using the points (-4, -4) and (8, 8), calculate the slope as m = (8 - (-4)) / (8 - (-4)) = 1.

Identify the y-intercept (b) by observing where the line crosses the y-axis. In this case, the line crosses the origin, so b = 0.

Write the equation of the line using the slope and y-intercept: y = 1x + 0, or simply y = x.

To find the area under the curve from x = -4 to x = 8, calculate the definite integral of the function y = x from x = -4 to x = 8. This involves finding the antiderivative of x, evaluating it at the upper and lower limits, and subtracting the results.