(a) The integrand of each definite integral is a difference of two functions. Sketch the

Chapter 9, Problem 139

(choose chapter or problem)

(a) The integrand of each 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.

\(\int_{0}^{1}(1-x) d x \quad \int_{0}^{1}\left(x-x^{2}\right) d x \quad \int_{0}^{1}\left(x^{2}-x^{3}\right) d x\)

(b) Find the area of each region in part (a).

(c) Let \(a_{n}=\int_{0}^{1}\left(x^{n-1}-x^{n}\right) d x\). Evaluate \(a_{n}\) and \(\sum_{n=1}^{\infty} a_{n}\). What do you observe?

Text Transcription:

int_0^1(1-x)dxint_0^1(x-x^2t)dxint_0^1(x^2-x^3)dx

a_n=int_0^1(x^n-1-x^n)dx

a_n

sum_n=1^inftya_n