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6:04{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.
ƒ(𝓍) = tan⁻¹ (3x - 1) on [0,1]
(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.
Area functions for linear functions Consider the following functions ƒ and real numbers a (see figure).
(b) Verify that A'(𝓍) = ƒ(𝓍).
ƒ(t) = 3t + 1 , a = 2
Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.
(b) ∫₃⁶ (―3g(𝓍)) d𝓍
Working with area functions Consider the function ƒ and the points a, b, and c.
(b) Graph ƒ and A.
ƒ(𝓍) = 1/𝓍 ; a = 1 , b = 4 , c = 6
Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.
(b) ∫₁⁶ (f(𝓍) ― g(𝓍)) d𝓍
{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.
∫₀⁴ (4𝓍― 𝓍²) d𝓍
