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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

Example 1

A snail travels with velocity v(t) = 2 + 0.5sin t feet per second, where t is given in seconds. If the snail starts traveling at noon (t = 0), what does the expression

represent in the context of this problem?

Example 2

Water pressure p at a depth of h feet below the surface of the water is given by the formula

p(h) = 62.4h

measured in "pcf," or "pounds per cubic foot."

Evaluate the integral

and explain what it means in terms of this problem.

Example 3

Sonya can paint at a rate of

v(t) = 150 – 4t

square feet per hour, where t is the number of hours since she started painting. How long will it take Sonya to paint 568 square feet?

Example 4

A pan of brownies has been prepared at room temperature, 70°F, and is put into an oven that has been pre-heated to 350°F. When the brownies have been in the oven for t minutes they heat up at a rate inversely proportional to (t + 1), with proportionality constant k. If the brownies are 250 degrees after being in the oven for 15 minutes, what is k?

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