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- Evaluate the limit limn→∞ ∑ n i=1 f(ci ) Δxi over the region bounded by the graphs of the equations.sketch the region and use the limit definition to find the area of region between the graph of f(x) = (1/2)x2 +1 and x-axis over the interval [-3,0]Find the area between the graph of y-5-5x^(2/3) and x-axis and −1≤ x ≤ 8 using the right hand rule of Riemann sums.
- 1. a) Find the area of the region lying between f(x) = x^2 − 3x + 2 and x − axis between x= 2and x = 4 by the limit definition. 1. b) Check your answer in a) by finding a definite integral.11.Find the area under the curve y=sin2xcosx for the given limits; sketch the area. Hint: integral in a solution has a format of general power formula. b) from x=π to x=3π/2Using the definition of integral12. x^2 dx as a limit of a sum of rectangles, compute integral12. x^2 dx. (Hint: You will likely need to use the facts that Sigmai=1n i^2= n(n+1)(2n+1)/6 and Sigmai=1n i= n(n+1)/2)
- Use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval; f(x)=2x3+x2+3, [0,3] using sigma notationEvaluate the limit lim xi over the region bounded by the graphs of the equations. f(x) = √x, y = 0, x = 0, x = 3Estimate the area of the region bound by the graph of f(x) = x^2 + 3x and the x-axis on [2,5] using a right-endpoint Riemann sum with n = 50 rectangles.
- In the given equation , use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval. Sketch the region y = 2x - x3 , [0, 1]Find the exact area of the region under the graph of y= e -x from 0 to 2 by using a computer algebra system to evaluatethe sum and then the limit in Example 3(a). Compare youranswer with the estimate obtained in Example 3(b).