Evaluate the triple integral ∫∫∫E3ydV,where E={(x,y,z)∣0≤x≤2,0≤y≤4−x2,0≤z≤x}

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Evaluate the triple integral ∫∫∫E3ydV,where  E={(x,y,z)∣0≤x≤2,0≤y≤4−x2,0≤z≤x}

Obiajulu · Accepted Answer

Hey there. So the region of integration E is a simple region. That means that it doesn&#039;t do anything wonky and that we can really just use those defining inequalities as is in our integral. So let&#039;s write out the integral as an iterated integral. Note that the order of y and z doesn&#039;t technically matter (though practically, one may be harder than the other), but the integral with respect to x needs to go on the outside because the other two inequalities are defined in terms of x. So let&#039;s set up the integral like this. ∫E3ydV=∫02∫04−x2∫0x3ydzdydx Now we just evaluate one integral at a time, starting on the inside. With respect to z, 3y is constant and hence =∫02∫04−x23y[z]0xdydx=∫02∫04−x23yxdydx With respect to y, the 3x is constant, but there is a y left to integrate. =∫023x[y22]04−x2dx=∫0232x|4−x2|dx Note that on the interval [0,2],the expression 4−x2 is always nonnegative, so I can remove that absolute value. And so, evaluating that last integral we get ... =6∫02xdx−32∫02x3dx=6

Evaluate the triple integral intintint_E3ydV,where E={(x,y,z)|0<=x<=2,0<=y<=sqrt(4-x^2),0<=z<=x}

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