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Chapter 4: Q. 40 (page 405)

Combining derivatives and integrals: Simplify each of the following as much as possible.

ddx鈭�0x鈥�t3dt

Short Answer

Expert verified

ddx鈭�0x鈥�t3dt=x3

Step by step solution

01

Step 1. Given information.

Given expression isddx鈭�0x鈥�t3dt

We have to simply solve the expression.

02

Step 2. Solve the expression.

If f is continuous on [a,b] then for all x鈭�[a,b]

ddx鈭�ax鈥�f(t)dt=f(x)

So

f(t)=t3f(x)=x3

And

ddx鈭�0x鈥�t3dt=x3

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Most popular questions from this chapter

Your calculator should be able to approximate the area between a graph and the x-axis. Determine how to do this on your particular calculator, and then, in Exercises 21鈥�26, use the method to approximate the signed area between the graph of each function f and the x-axis on the given interval [a, b].

f(x)=1-2x,[a,b]=[-3,1]

Consider the region between f and g on [0, 4] as in the

graph next at the left. (a) Draw the rectangles of the left-

sum approximation for the area of this region, with n = 8.

Then (b) express the area of the region with definite

integrals that do not involve absolute values.

Use the graph of f to estimate the values of A(1), A(2), A(3)

If 鈭�-23f(x)dx=4,鈭�-26f(x)dx=9,鈭�-23g(x)dx=2, and 鈭�36g(x)dx=3, then find the values of each definite integral in Exercises 29-40. If there is not enough information, explain why.

鈭�-2-2x(f(x)+3)2dx

Write out all the integration formulas and rules that we know at this point.
See all solutions

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