Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

9. Graphical Applications of Integrals

Area Between Curves

Calculus

9. Graphical Applications of Integrals

Area Between Curves

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Multiple Choice

Which of the following integrals correctly represents the area of the region enclosed by the curves $y = 2 x$ , $y = 8 x$ , and $y = 18 x$ for $x > 0$ ?

\int_{x = 0}^{x = 1} (18 x - 2 x) d x

\int_{x = 0}^{x = 1} (18 x - 8 x) d x

\int_{x = 0}^{x = 1} (8 x - 2 x) d x

\int_{x = 0}^{x = 1} (8 x - 2 x) d x

\int_{x = 0}^{x = 1} (18 x - 8 x) d x

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Verified step by step guidance

Step 1: Understand the problem. The goal is to find the area of the region enclosed by the curves y = 2x, y = 8x, and y = 18x for x > 0. This involves determining the correct integral representation for the area between these curves.

Step 2: Analyze the curves. For x > 0, the curves y = 2x, y = 8x, and y = 18x are linear functions with increasing slopes. The curve y = 2x is the lowest, y = 8x is in the middle, and y = 18x is the highest.

Step 3: Break the region into subregions. The area enclosed by these curves can be split into two parts: (1) the area between y = 8x and y = 2x, and (2) the area between y = 18x and y = 8x.

Step 4: Write the integral expressions for each subregion. For the area between y = 8x and y = 2x, the integral is $ \int_{0}^{1} (8x - 2x) \, dx $. For the area between y = 18x and y = 8x, the integral is $ \int_{0}^{1} (18x - 8x) \, dx $.

Step 5: Combine the integrals. The total area is the sum of the two integrals: $ \int_{0}^{1} (8x - 2x) \, dx + \int_{0}^{1} (18x - 8x) \, dx $. This represents the correct integral expression for the area of the region enclosed by the curves.