Sketch a continuous function ƒ that is positive over the interval [0, 1) and negative over the interval (1, 2], such that the net area of the region bounded by the graph of ƒ and the x-axis on [0, 2] is zero.

Q: Plot the functions y1 : R → R, y2 : R → R given by y1(x) = x2 − x and y2(x) = x. Compute the area…

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Q: Find the area betweeen y=x2 and y= -(x-1)2+1 on the interval [0,2]

A: Given: Equations of curves are y=x2 y= -(x-1)2+1 to find the area on the interval [0, 2] The area…

Q: find the area between f(x)=x2+2 and g(x)=-x+1 on [1,3]

A: Draw the graph. To find the region between the red and blue function: Red: y=x2+2 Blue: y=-x+1…

Q: What would be an example of a function that is not integrable on [2, 10] and why is it not…

A: Consider a function: y=1x-2……1

Q: Find-the-approximate area-of-the-region-bounded-by-the-curves y = x/Vx2 + 1•-and-y x4 - x1 %3D

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Q: Find the absolute extrema of the function f defined by f(x) triangular region R with vertices (0,…

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Q: Use right hand rule to find area under the curve   f(x)= x2-x on interval [1,3]

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Q: Find the points on the graph of the function f(x) = x^2 that are closest to (0, 2) in the xy-plane.…

A: Minimal distance

Q: Find the absolute maxima and minima of f(x,y) on the given region: f(x,y)= x^3 - y^3 - 27xy…

A: Solution :-

Q: Approximate the net signed area under the curve y = x 2 − 2x − 2 on the interval [1, 5] using 4…

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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.