Merit Worksheet #1, 1/21/09
Let’s talk about integrals. As a group, answer the following.
1. Shown below is a picture of f(x) on the interval [1,3].
(a) Conceptually, what exactly does Z3
1
f(x)dx represent?
(b) What would Z3
1
f(x)dx be, in this case?
2. Find the following quantities.
(a) The area between y=x2and the x-axis on the interval [0,1].
(b) Z−1
−2
1
xdx +Z1
0
1
1 + x2dx −Zπ/2
0
sin x dx.
(c) R1
−1
1
√1−x2dx +R1
0exdx.
3. (a) Show that F1(x) = 1/(1 −x) is an antiderivative of f(x) = 1/(1 −x)2.
(b) Show that F1(x) = x/(1 −x) is also an antiderivative of f(x)=1/(1 −x)2.
(c) How is it possible that these two different functions are both antiderivatives of f(x)? What, if
any, is the relationship between the two?
4. Shown below is the graph y=√1−x2on the interval [−1,1].
(a) What’s Z1
−1p1−x2dx?
(b) What makes finding Z1/2
−1p1−x2dx tricky (for you, who’ve only seen Calculus I)?
(c) Using what you learned in Calc I, how could you approximate Z1/2
−1p1−x2dx?
(d) Wonder aloud (with feeling) whether it’s possible that you’ll learn techniques in Calc II allowing
you to find Rb
a√1−x2dx for any aand b?
(e) Throw calculus aside and use geometry (and maybe some trigonometry) to calculate Z1/2
−1p1−x2dx.
