Sketch the region enclosed by x + y² = 42 and r + y = 0. Decide whether to integrate with respect to æ or y. Then find the area of the region. %3D
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.
Given curve is:
Line is:
The region enclosed by these two is shown in the figure below:
The area of the shaded region is same as the area of the region of the curve from towards positive X-axis, due to the symmetry of the region to be excluded and included are same.
Integration with respect to y of the curve is:
To integrate with respect to y the above expression, find the limits of y along the axis:
The point where curve crosses Y-axis is:
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