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07:33{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.
(b) Calculate g'(π)
g(π) = β«βΛ£ sin (ΟtΒ² ) dt ( a Fresnel integral)
Properties of integrals Use only the fact that β«ββ΄ 3π (4 βπ) dπ = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(b) β«ββ΄ π(π β 4) d(π)
{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.
f(x) = sin 2x on [0,3Ο/4]
(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.
The following functions are positive and negative on the given interval.
Ζ(π) = xeβ»Λ£ on [-1,1]
(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.
{Use of Tech} Riemann sums for larger values of n Complete the following steps for the given function f and interval.
Ζ(π) = xΒ² β 1 on [2,5] ; n = 75
(b) Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f and the x-axis on the interval.
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(b) A left Riemann sum always overestimates the area of a region bounded by a positive increasing function and the x-axis on an interval [a,b].
