Final answer:
The student is asking about calculating the area between the curves x + y = 13 and x + 7 = y² using integrals. The area can be found by setting up integrals with respect to the x-axis or y-axis after determining the intersection points. This method involves subtraction of one curve from the other within the bounds of the established limits.
Step-by-step explanation:
The question pertains to the computation of the area between two curves, represented by the equations x + y = 13 and x + 7 = y². To find this area using integrals, we can set up our integral(s) with respect to either the x-axis or the y-axis. When setting up the integrals with respect to the x-axis, we'll look for the points where the two curves intersect, and integrate y as a function of x between these two points, subtracting the lower curve from the upper one. Alternatively, by solving for x as a function of y, we can integrate with respect to the y-axis, again subtracting the curve on the left from the curve on the right. This method may involve integrating over two separate intervals if the curves intersect the y-axis in more than one place.
To proceed, we first must rearrange each equation and solve for the points of intersection. The first equation, when solved for y, becomes y = 13 - x. The second equation represents a parabola that opens upwards with its vertex at point (0, -7) on the y-axis, and it can be rewritten as x = y² - 7.
Setting y = 13 - x equal to y = √(x + 7), we find the values of x for which the curves intersect. These points define the limits of integration for the x-integrals. Similarly, by setting x = y² - 7 equal to x = 13 - y, we find the values of y for which the curves intersect, establishing the limits for the y-integrals.
The calculation of these integrals will reveal the exact area between the curves, which can be interpreted physically in various contexts, such as the work done by a force over a displacement (as suggested by Figure 7.9) or the displacement in a velocity versus time graph.