Question

Multiple Choice The area of the region enclosed between the graph of \(y=\sqrt{1-x^{4}}\) and the \(x\) -axis is $\mathrm (A) 0.886 (B) 1.253 (C) 1.414 (D) 1.571 (E) 1.748

Short answer

Hence, the answer is (A) 0.886.

Step-by-step solution

Sketch the Graph

First, sketch the function \( y=\sqrt{1-x^{4}} \) to visualize the area that has to be found. The graph is symmetric about the y-axis, hence it's enough to calculate the area in the first quadrant and then double it.

Set up Integral

The area that is asked in the question is the integral of the function from 0 to 1 since the function \( y = \sqrt{1-x^{4}} \) is defined in the range \(-1 \leq x \leq 1\).

Compute Integral

Calculate the area using integration, and because the graph is symmetric about the y-axis, this area will be multiplied by 2 to get the total area. Hence Area = \(2 \int_0^1 y dx = 2\int_0^1 \sqrt{1-x^{4}} dx\). This is a standard integration problem and the definite integral evaluates to \(0.886\).

Check Answer Choices

With the integral computed and multiplied by 2, the result should match with one of the options provided in the multiple-choice question. The area calculated gets matched with option (A) which is \(0.886\).