# CLEAR Calculus

## Lab 17: The fundamental theorem of calculus - part 3

The following diagram represents eight different ways you might use the Fundamental Theorem of Calculus. For example, #7 represents using a table of values for a function to compute values for its antiderivative. Classify each of the following problems as 1-8 from the diagram then solve the problem.

Estimate$$\int_0^{100} f(t) dt$$ given the following table.

 $$t$$ 0 20 40 60 80 100 $$f(t)$$ 1.2 2.8 4 4.7 5.1 5.2 Find $$F(-1.5)$$ given the graph of $$f(x) = F'(x)$$ and $$F(1) = 0$$ shown to the right.

If $$f'(x) = e^{-x^2}$$ and $$f(0) = 2$$ find $$f(1)$$ and $$f(-1)$$.

Find the area below the graph of $$y = \cos x^2$$ from $$x = 0$$ to $$x = \sqrt{\frac{\pi}{2} }$$.

Find $$\int_{-4}^4 f(x)dx$$ given the graph shown to the right. Find $$\int_0^1 3e^{-2x} dx$$.

Find the area between the graphs of $$y = x^2$$ and $$y = 2x^2 + x - 6$$.

If $$f(0) = 100$$ estimate $$f(x)$$ for $$x =2,4,$$ and $$6$$ given the following table.

 $$x$$ 0 2 4 6 $$f'(x)$$ 10 18 23 25