We can estimate the average value of a region of level curves by using the formula (1/A(R)) int int_R f(x,y) Delta(A), where A(R) is the area of the rectangle defined by R=[x1,x2]x[y1,y2], and where the double integral gives the volume under the surface f(x,y) over the region R.
Read MoreThe Mean Value Theorem for integrals tells us that, for a continuous function f(x), there’s at least one point c inside the interval [a,b] at which the value of the function will be equal to the average value of the function over that interval. This means we can equate the average value of the function over the interval to the value of the function at the single point.
Read MoreThink about the average value of a function as the average height the function attains above the x-axis. If the function were y=3, then the height of the function is always 3 everywhere, so the average height of the function would also be 3. When the function gets more complicated, we can use the average value formula to find its average height on [a,b].
Read MoreIn order to use the triple integral average value formula, we’ll have find the volume of the object, plus the domain of x, y, and z so that we can set limits of integration, turn the triple integral into an iterated integral, and replace dV with dzdydx.
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