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Limit definition of the definite
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- Use the limit definitionof the definite integral with right Riemann sums and a regular partitionto evaluate the following definite integrals. Use the FundamentalTheorem of Calculus to check your answer.arrow_forwardPython: Simpson's rule says the integral from x_0 to x_2 of f(x)dx is approximately h(1/3 f(x_0) + 4/3 f(x_1) _1/3 f(x_2)) where h = x_2-x_0 and x_1 is the midpoint of x_0 and x_2. Write a function simp(f,a,b,n) which integrates the function f(x) over the interval [a,b] by dividing it into n subintervals. integrate e^-x over [0,1] to make sure it matches the integral found with scipy.integrate.quad to 5 decimal placesarrow_forwardFind the area (in square units) of the region under the graph of the function f on the interval [1, 8], using the Fundamental Theorem of Calculus. Then verify your result using geometry. f(x) = -(1/8)x+1arrow_forward
- Find an area similar to the example f(x)=x^2 from [0,1] that we couldn't have computed from geometry. Use Riemann sums to calculate (limits, etc). Area cannot simply be a rectangle, triangle or a circle (as this is a calculus problem). Give a detailed explanation and show all work.arrow_forwardRegion R in the image is bounded by P1: y=2 , P2: y=1+(3/x), and P3: 2y+x=9. Set up and do not evaluate a (sum of) definite integral(s) which represent the following:1. the area of R, using horizontal strips 2. the region's volume it is revolved about y = 1, using the method of washersarrow_forwardDecided whether the integral is improper. Explain your reasoning. 5 −5 ln(x16) dx The integral is improper because the upper limit of integration is infinite.The integral is improper because the lower limit is outside the domain of the function. The integral is improper because both the upper limit of integration and the lower limit of integration are infinite.The integral is proper.The integral is improper because the function has an infinite discontinuity in [−5, 5].arrow_forward
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