8. Definite Integrals

Let $g\left(x\right)={\int }_0^{x}f\left(t\right) dt$, where $f$ is the function whose graph is shown. Which of the following statements is true about $g^{\prime }\left(x\right)$?

Evaluate the following derivatives.

d/d𝓍 ∫₃ᵉˣ cos t² dt

For Exercises 127 and 128 find a function f satisfying each equation.

128. f(x) = e² + ∫₁ˣ f(t) dt

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

The linear function ƒ(𝓍) = 3 ― 𝓍 is decreasing on the interval [0, 3]. Is its area function for ƒ (with left endpoint 0) increasing or decreasing on the interval [0, 3]? Draw a picture and explain. 

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(c) ∫ₐᵇ ƒ'(𝓍) d𝓍 = ƒ(b) ―ƒ(a) .

Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Explain why your result is consistent with the figure.

∫₀¹ (𝓍² ― 2𝓍 + 3) d𝓍

Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.

15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫₋₂⁻¹ 𝓍⁻³ d𝓍

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.

37. ∫(from x²/2 to x²)ln(√t)dt

Working with area functions Consider the function ƒ and the points a, b, and c.

(a) Find the area function A (𝓍) = ∫ₐˣ ƒ(t) dt using the Fundamental Theorem.

ƒ(𝓍) = ― 12𝓍 (𝓍―1) (𝓍― 2) ; a = 0 , b = 1 , c = 2

Differentiating Integrals

In Exercises 75–78, find dy/dx.

________

y = ∫₂ˣ √ 2 + cos³t dt

Suppose F is an antiderivative of ƒ and A is an area function of ƒ. What is the relationship between F and A?