Question

In Exercises \(15-34,\) find the area of the regions enclosed by the lines and curves. $$4 x^{2}+y=4 \quad$$ and $\quad x^{4}-y=1$$

Short answer

Without calculating the integral, the exact value cannot be determined. However, the integral set up in the last step provides the formula to compute the numerical value for the area enclosed by the two curves.

Step-by-step solution

Identify Points of Intersection

To find these points, set \(4x^{2}+y\) equal to \(x^{4}- y\): \[4x^{2}+y = x^{4}- y\] Solving the above equation gives \(x^{4} - 4x^{2} - 2y = 0\]. Express this equation in terms of y to find the y-values of intersection points.

Solve the Equation for y

Express the equation in terms of y: \[2y = x^{4} - 4x^{2}\] So, \[y = \frac{1}{2} x^{4} - 2x^{2}\]

Find the Area Between the Curves

Calculate the area between the curves by integrating over the interval defined by the intersection points. To find the definite integral for the area, subtract the smaller function from the larger one and integrate over the real interval. It can be proven that the smaller function is \(y = \frac{1}{2} x^{4} - 2x^{2}\) and the larger is \(y = 4 - 4x^{2}\). Therefore the integral to calculate the area \(A\) is as follows: \[ A = \int_{-a}^{a} [(4 - 4x^{2}) - (\frac{1}{2} x^{4} - 2x^{2})] dx\], where 'a' is the x-coordinate of the points of intersection.

Compute the Integral

Calculate the definite integral to get the area. This requires further steps of computing the integral including applying the power rule, applying the limits of integration, and subtracting.