Problem 55 - Essential Calculus

Question

(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.

Step-by-Step Solution

Answer

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 1: 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 \(\int_{0}^{1} \arcsin x dx\).

Step 2: 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.

Step 3: 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.