Final answer:
To sketch the region enclosed by the given curves, plot the curves and find their points of intersection. Integrate with respect to the variable that changes more rapidly. Draw a typical approximating rectangle by choosing intervals of y-values and finding the corresponding x-values.
Step-by-step explanation:
To sketch the region enclosed by the given curves, we need to find their points of intersection. Setting the equations equal to each other, we get x²−4x = 2x+7. Rearranging this equation, we have x²−6x-7=0. Solving for x using the quadratic formula, we find x = 7 and x = -1. Next, we can plot these two points on a graph and draw the curves y = x²−4x and y = 2x + 7. The region enclosed by the curves will be between these two curves from x = -1 to x = 7.
To decide whether to integrate with respect to x or y, we look at the y-values of the intersection points. The y-values are y = (-1)² - 4(-1) = 5 and y = 2(7) + 7 = 21. Since the y-values are changing more rapidly, we will integrate with respect to y.
To draw a typical approximating rectangle, we can choose a small interval of y-values and find the corresponding range of x-values using each curve equation. Then, we can draw a rectangle that has a height of the y-interval and a width of the x-interval. Repeat this process for multiple intervals to get a more accurate approximation of the enclosed region.