Use a triple integral to find the volume of the region bounded by y = x^2, z = 0, and y + z = 1 in the order dz dy dx
-
Volume = ∫[-1, 1] ∫[0, x^2] ∫[0, 1 - y] dz dy dx
= ∫[-1, 1] ∫[0, x^2] (1 - y) dy dx
= ∫[-1, 1] x^2 - (x^4)/2 dx
= (x^3)/3 - (x^5)/10 eval. from -1 to 1
= 1/3 - 1/10 + 1/3 - 1/10 = 2/3 - 1/5 = 7/15
= ∫[-1, 1] ∫[0, x^2] (1 - y) dy dx
= ∫[-1, 1] x^2 - (x^4)/2 dx
= (x^3)/3 - (x^5)/10 eval. from -1 to 1
= 1/3 - 1/10 + 1/3 - 1/10 = 2/3 - 1/5 = 7/15