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Chapter 4: Problem 55

(a) Sketch the region whose area is represented by \(\int_{0}^{1} \arcsin x d x\) (b) Use the integration capabilities of a graphing utility to approximate the area. (c) Find the exact area analytically.

Short Answer

Expert verified
By using the analytical solution, the exact area under the curve \(y = \arcsin x\), for \(x\) between 0 and 1, is found to be \(\frac{\pi}{4} - \frac{1}{2}\).

Step by step solution

01

Drawing of the graph

First, on a Cartesian plane, graph the function \(y = \arcsin x\) for \(x\) ranging from 0 to 1. The area under the curve of this function above the \(x\)-axis between these two \(x\)-values corresponds to the value of the integral \(\int_{0}^{1} \arcsin x dx\).
02

Approximate area calculation

Load the graph of the function \(y = \arcsin x\) onto a graphing calculator, and use its function to calculate numerical definite integrals. Set the upper and lower limits of integration to be 1 and 0 respectively to get the approximate value of the integral \(\int_{0}^{1} \arcsin x dx\), which is an estimation of the area under the curve.
03

Analytical Calculation

To determine the exact area, use integration methods. Since the integral of \(\arcsin x\) is \(x\arcsin x + \sqrt{1 - x^2}\), compute \(\int_{0}^{1} \arcsin x dx\) from these antiderivative and the limits of integration. At the end, the exact value obtained will be the exact area under the curve.

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Most popular questions from this chapter

(a) integrate to find \(F\) as a function of \(x\) and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). $$ F(x)=\int_{4}^{x} \sqrt{t} d t $$

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