Question

Find the area of the region enclosed by the two functions y=5x^2 and y+x^2+4.

Answer 1

The area of the region enclosed by the functions
`\(y = 5x^2\)`and
`\(y = x^2 + 4\) is \(8.4\)`square units.

Explanation:

To find the area between two functions, determine their intersection points by setting them equal to each other:
`\(5x^2 = x^2 + 4\).` Simplify to get
`\(4x^2 = 4\) and \(x^2 = 1\),` which leads to
`\(x = \pm 1\).` Integrating
`(y = 5x^2\)` and \(y = x^2 + 4\) with respect to
`\(x\) from \(-1\) to \(1\),`calculate the definite integral of
`\(5x^2 - (x^2 + 4)\`) within these bounds.

The integral becomes
`\(\int_(-1)^(1) (5x^2 - x^2 - 4) dx\),`which simplifies to
`\(\int_(-1)^(1) 4x^2 - 4 dx\).` Evaluate this integral to find the area enclosed, resulting in 8.4 square units.

This solution involves identifying the intersection points, setting up the integral bounds using these points, subtracting the functions, and integrating the absolute difference between them within those bounds to determine the area.