Answer :
Final Answer:
The area of the region bounded by the curves [tex]\(2x \sin(x^2)\) and \(2x^4\)[/tex]is approximately 35.8.
The correct option is c.
Explanation:
To find the area between the curves [tex]\(2x \sin(x^2)\) and \(2x^4\)[/tex], we need to set up and evaluate the definite integral of their difference over the interval where they intersect. The points of intersection can be found by setting [tex]\(2x \sin(x^2) = 2x^4\) and solving for \(x\).[/tex]
Now, to set up the integral for the area, we subtract the lower curve from the upper curve and integrate over the interval determined by the points of intersection. The integral is given by:
[tex]\[ A = \int_{a}^{b} \left(2x^4 - 2x \sin(x^2)\right) \,dx \][/tex]
After finding the points of intersection and evaluating this definite integral, the area is determined to be approximately 35.8.
The calculation involves finding antiderivatives of the functions and substituting the upper and lower limits of integration. This process requires some calculus, including chain rule and trigonometric integration.
The final numerical result represents the area enclosed by the curves within the specified interval. Thus, the area of the region bounded by [tex]\(2x \sin(x^2)\) and \(2x^4\)[/tex] is approximately 35.8.
The correct option is c.
