Question

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{2}^{3}\left[\left(\frac{x^{3}}{3}-x\right)-\frac{x}{3}\right] d x $$

Name: Problem 10, Chapter 7: Calculus
Uploaded: 2019-07-25T00:25:47-08:00
Duration: 10 s
Channel: Sahil Patel
Description: The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. ∫2^3[((x^3)/(3)-x)-(x)/(3)] d x

   The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral.
$$
\int_{2}^{3}\left[\left(\frac{x^{3}}{3}-x\right)-\frac{x}{3}\right] d x
$$

Calculus

Ron Larson, Robert… 8th Edition

Chapter 7, Problem 10

Instant Answer

Solved by Expert Sahil Patel

07/25/2019

Step 1

The integrand is given as $\left(\frac{x^{3}}{3}-x\right)-\frac{x}{3}$. We can combine the terms: \[ \frac{x^{3}}{3} - x - \frac{x}{3} = \frac{x^{3}}{3} - \frac{3x}{3} - \frac{x}{3} = \frac{x^{3}}{3} - \frac{4x}{3} \] Thus, the integrand simplifies to Show more…

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The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{2}^{3}\left[\left(\frac{x^{3}}{3}-x\right)-\frac{x}{3}\right] d x $$

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Key Concepts

Definite Integral

A definite integral computes the accumulation of a quantity, such as area, over a specific interval. It is defined by the limits of integration and includes the net contribution of the function values between these limits.

Functions and Graphing

Sketching the graphs of functions involves plotting their behavior over an interval, including key features like intercepts, slopes, and curvature. This visual representation aids in understanding the relationships between functions and is essential for identifying regions of interest.

Area Between Curves

The area between curves is found by taking the difference of the functions and integrating this difference over a given interval. This concept illustrates the method of calculating the region that lies between the graphs of two functions, as represented by a definite integral.

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