CALC 2 University of Toronto Gaussian Integral & Calculus Equations Homework Question1. Considerthesphereofradius : ithasequation 2+ 2+ 2 = 2. • Compute

CALC 2 University of Toronto Gaussian Integral & Calculus Equations Homework Question1. Considerthesphereofradius : ithasequation 2+ 2+ 2 = 2. • Computethesurfaceareaoftheportionoftheupperhalf-spherethatliesabove thedisk 2+ 2 = 2,where0 < < . • Take the limit of your answer as → − to find the area of the upper half sphere. • Deducetheareaofthewholesphere. Question2. Sketchthesolidwhosevolumeisgivenbytheiteratedintegral ∫ 2 0 ∫ 2− 0 ∫ 4− 2 0 1d d d . Question3. Let > 0 be a positive constant. Calculate∫∞ −∞ e− 2 d . Note: I am more interested in your work than in the answer. Hints: Do not re-calculate the entire Gaussianintegralforthisscaledintegrand. Weknowthat ∫ 0 −∞ e− 2 d +∫∞ 0 e− 2 d =∫∞ −∞ e− 2 d , andwehavealreadyfoundthevalueofthisintegral. Anotherhint: write∫∞ −∞ e− 2 d =∫ 0 −∞ e− 2 d +∫∞ 0 e− 2 d . 1 Question 1. Consider the sphere of radius a: it has equation x2 + y2 + z2 = a-.
• Compute the surface area of the portion of the upper half-sphere that lies above
the disk x² + y2 = 12, where 0
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