Volume of the solid (when squares divided into 4 equal parts in the lower left hand corners) = 318 cubic units.
Step-by-step explanation:
Paraboloid z=81-x2 -y2
The lower left hand corners of the squares are(0,0)(1,0)(0,1) and (1,1).
Then the volume can be estimated as,
Volume ≈ΔA [f(0,0)+f(1,0)+f(0,1)+f(1,1)]
= 1[81+79+80+78]=318 cubic units.
b.) estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the upper right hand corners.
Volume of the solid (when squares divided into 4 equal parts in the upper right hand corners) = 294 cubic units.
Paraboloid z=81-x2 -y2
The upper right hand corners of the squares are(1,1)(2,1)(1,2) and (2,2).
Then the volume can be estimated as,
Volume ≈ΔA [f(1,1)+f(2,1)+f(1,2)+f(2,2)]
= 1(78+69+75+72)=294 cubic units
c.) what is the average of the two answers from a and b
Average of part a and b answer = (318+294)/2 = 712/2=306 cubic units
d.) using iterated integrals compute the exact value of the volume.
0202(81-x2 -y2) dy dx = 02[81y-x2y-⅓ y3]02 dx
= 02[162-2x2-8/3] dx
= [162x-2x3/3-8/3 x]20
=644-16/3-16/3
= 644 cubic units.
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