4 4-y vz JI f (x,y,z)dxdzdy JI f(x,y,z)dydzdx 0 0 0 E
Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
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![The image shows a mathematical expression representing a triple integral and its variables of integration:
\[
\int_{0}^{4} \int_{0}^{4-y} \int_{0}^{\sqrt{z}} f(x, y, z) \, dx \, dz \, dy = \int \int \int_{E} f(x, y, z) \, dy \, dz \, dx
\]
- On the left side, we have a triple integral with specified limits where:
- The outermost integral with respect to \( y \) goes from 0 to 4.
- The middle integral with respect to \( z \) ranges from 0 to \( 4 - y \).
- The innermost integral with respect to \( x \) ranges from 0 to \( \sqrt{z} \).
- The right side of the equation represents the same triple integral over a region \( E \) with the order of integration \( dy \, dz \, dx \).
This expression is useful for evaluating the integral of a function \( f(x, y, z) \) over a described region by changing the order of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa91837df-a594-491b-baa0-532b65fd0256%2F4adfea59-85a3-449c-b24d-392d6f8b7f0f%2Fwr84h7s_processed.png&w=3840&q=75)
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