Question

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(y)=y^{2}+1, g(y)=4-2 y $$

Short answer

Calculate the integral after finding out the correct functions' order between \(y_1\) and \(y_2\). This solution gives the area between the two curves. Final answer will be provided upon completing calculations.

Step-by-step solution

Find intersection points

Set the two functions equal to each other and solve for y: \(y^{2}+1 = 4-2y\) . This gives a quadratic equation: \(y^{2} + 2y - 3 = 0\), which can be solved to obtain the intersection points \(y_1\) and \(y_2\).

Determine functions order

It's crucial to determine which function is on top and which function is on the bottom within the integration interval. To find out, pick any point \(y\) between \(y_1\) and \(y_2\) and check the value of each function at this point, thus determining which function is greater.

Set up and compute the integral

For any given interval \([y_1, y_2]\), the area \(A\) between the two curves \(f(y)\) and \(g(y)\) is given by \(A = \int_{y_1}^{y_2} [g(y) - f(y)] dy\). Substitute the explicit forms of the functions f and g into this formula, and then perform the integral calculation to find the area.