Set up, but do not evaluate, an integral or integrals for the area bounded by the following curves. Do not use the absolute value function y = 8x and y = x³ – x
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 curves , y=8x and y=x3-x
we have to set up the integral for the area bounded by the following curves.
we know that,
The Area Under a Curve
The area under a curve between two points can be found by doing a definite integral between the two points.
To find the area under the curve y = f(x) between x = a and x = b, integrate y = f(x) between the limits of a and b.
firstly , solve the both curves to get the point of intersection,
8x=x3-x
x3-x-8x=0 x(x2-9)=0 x(x-3)(x+3)=0
x=0,3,-3 is the point of intersection of the curves
here the graph of both curves is
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