Math 152 Workshop 2
Jan 27, 2021
Exercise 2.1 Let R be the region in the xy-plane bounded between the parabola y = 1 − x2 and the x-axis. An architect is assigned a
project to design several different buildings using the region R as the foundation. The firm’s client wants to see specifications for each of
these building prototypes, since the final decision on the building prototype will take into account the aesthetics and the costs associated
with the specifications.
(a) The architect needs a “concrete” calculation (pun intended!) of the area of the foundation. Use a definite integral to find the area of
R.
(b) It is really important that the area calculation is correct. The architect wants to double-check the area calculation by using a different
definite integral to verify the value of the first integration. What could s/he use? Set-up and integrate this second definite integral
that computes the area of R.
(c) The client has asked the architect to put together 6 different designs. The first three designs use cross-sections perpendicular to the
x-axis to construct the buildings. Find the volume of the building if the cross-sections perpendicular to the x-axis are
(i) squares whose base lies in R
(ii) semi-circles whose diameter lies in R
(iii) equilateral triangles whose base lies in R
(d) The next three designs use cross-sections perpendicular to the y-axis to construct the buildings. Find the volume of the building
created if the cross-sections perpendicular to the y-axis are
(i) squares whose base lies in R
(ii) semi-circles whose diameter lies in R
(iii) equilateral triangles whose base lies in R
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