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

Given the region bounded above by $y = x^{2}$ and below by $y = x$ on the interval $0 \leq x \leq 1$ , determine the $x$ - and $y$ -coordinates of the centroid of the shaded area.

x = 0.7

y = 0.25

x = 0.6

y = 0.35

x = 0.583

y = 0.283

x = 0.5

y = 0.5

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Verified step by step guidance

Step 1: Understand the problem. The centroid of a region is the 'center of mass' of the area. To find the centroid, we need to calculate the x-coordinate (denoted as $ \bar{x} $) and y-coordinate (denoted as $ \bar{y} $) using formulas for the centroid of a region bounded by curves.

Step 2: Recall the formulas for the centroid. The x-coordinate of the centroid is given by $ \bar{x} = \frac{1}{A} \int_{a}^{b} x \cdot [f(x) - g(x)] \, dx $, and the y-coordinate is given by $ \bar{y} = \frac{1}{A} \int_{a}^{b} \frac{[f(x) + g(x)]}{2} \cdot [f(x) - g(x)] \, dx $, where $ A $ is the area of the region, $ f(x) $ is the upper curve, and $ g(x) $ is the lower curve.

Step 3: Calculate the area $ A $ of the region. The area is given by $ A = \int_{a}^{b} [f(x) - g(x)] \, dx $. Here, $ f(x) = x^2 $ and $ g(x) = x $, and the interval is $ [0, 1] $. Set up the integral $ A = \int_{0}^{1} (x^2 - x) \, dx $. Evaluate this integral to find $ A $.

Step 4: Set up the integral for $ \bar{x} $. Substitute $ f(x) = x^2 $ and $ g(x) = x $ into the formula for $ \bar{x} $: $ \bar{x} = \frac{1}{A} \int_{0}^{1} x \cdot (x^2 - x) \, dx $. Simplify the integrand and evaluate the integral.

Step 5: Set up the integral for $ \bar{y} $. Substitute $ f(x) = x^2 $ and $ g(x) = x $ into the formula for $ \bar{y} $: $ \bar{y} = \frac{1}{A} \int_{0}^{1} \frac{(x^2 + x)}{2} \cdot (x^2 - x) \, dx $. Simplify the integrand and evaluate the integral. Combine the results to find the coordinates of the centroid.