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Chapter 6: Problem 33

Evaluate the definite integral. Use a graphing utility to confirm your result. $$ \int_{0}^{1 / 2} \arccos x d x $$

Short Answer

Expert verified
The solution requires using integration by parts and substitution of the integral limits. The details of the solution are shown in the steps. The final result should be confirmed using a graphing utility, ensuring the area under the curve of the \( \arccos x \) function matches the calculated result.

Step by step solution

01

Setting up the integration by parts

Integration by parts follows the formula: \( \int u dv = uv - \int v du \). Set \( u = \arccos x \) and \( dv = dx \). Then differentiate and integrate to get \( du = -\frac{1}{\sqrt{1-x^2}}dx \) and \( v = x \).
02

Apply the integration by parts formula

Now we can apply the integration by parts formula: \( \int u dv = uv - \int v du \). This gives us:\[ x\arccos(x) - \int -x / \sqrt{1-x^2} dx \]
03

Resolve the remaining integral

The remaining integral \( \int -x/\sqrt{1-x^2} dx \) is a standard integral in the form: \( \int x\sqrt{a^2 - x^2} dx \) which results in \(-\frac{1}{3}*\sqrt{1-x^2}^3 \). Now we rewrite the original integral as:\[ x\arccos(x) + \frac{1}{3}\sqrt{1-x^2}^3 \].
04

Evaluate the definite integral

Substitute the limits of the definite integral (0 and 1/2) into the expression \[ x\arccos(x) + \frac{1}{3}\sqrt{1-x^2}^3 \]to get the final result.
05

Confirm the result graphically

Using a graphing calculator or software, sketch the graph of \( y = \arccos x \), and check that the area under the curve from \( x = 0 \) to \( x = 1/2 \) corresponds to the value obtained in Step 4. This will confirm the result.

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Most popular questions from this chapter

Describe the different types of improper integrals

True or False? In Exercises 67-70, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(p(x)\) is a polynomial, then \(\lim _{x \rightarrow \infty}\left[p(x) / e^{x}\right]=0\).

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is continuous on \([0, \infty)\) and \(\lim _{x \rightarrow \infty} f(x)=0\), then \(\int_{0}^{\infty} f(x) d x\) converges

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is continuous on \([0, \infty)\) and \(\int_{0}^{\infty} f(x) d x\) diverges, then \(\lim _{x \rightarrow \infty} f(x) \neq 0\)

Consider the limit \(\lim _{x \rightarrow 0^{+}}(-x \ln x)\) (a) Describe the type of indeterminate form that is obtained by direct substitution. (b) Evaluate the limit. (c) Use a graphing utility to verify the result of part (b). FOR FURTHER INFORMATION For a geometric approach to this exercise, see the article "A Geometric Proof of \(\lim _{l \rightarrow 0^{+}}(-d \ln d)=0\) " by John H. Mathews in the College Mathematics Journal. To view this article, go to the website www.matharticles.com.
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