Math 152 Midterm 1 Review Notes

Math 152 Midterm 1 Review Sheet (Spring 2025)

Disclaimers

• The review problems are designed to help you study for the exam.
• Everything discussed in lecture, recitation, or textbook sections covered is fair game for the exam.
• This includes material standard to Math 151 or pre-calculus.
• Individual lecturer's guidance takes precedence over this sheet if there are contradictions.

Exam Rules

• The exam is closed book.
• No calculators or devices are allowed.
• No formula sheet is provided.

Things to have memorized

• It's expected that you know basic facts such as $sin(π/6) = 1/2$ and $arctan(1) = π/4$.

Basic Antiderivatives

1. $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, $(n \neq -1)$
2. $\int \frac{1}{x} dx = ln |x| + C$
3. $\int e^x dx = e^x + C$
4. $\int a^x dx = \frac{a^x}{ln(a)} + C$, (a > 0, a \neq 1)
5. $\int cos(x) dx = sin(x) + C$
6. $\int sin(x) dx = -cos(x) + C$
7. $\int sec(x) dx = ln |sec(x) + tan(x)| + C$
8. $\int csc(x) dx = - ln | csc(x) + cot(x)| + C$
9. $\int tan(x) dx = ln |sec(x)| + C$
10. $\int cot(x) dx = ln |sin(x)| + C$
11. $\int sec^2(x) dx = tan(x) + C$
12. $\int csc^2(x) dx = - cot(x) + C$
13. $\int sec(x) tan(x) dx = sec(x) + C$
14. $\int csc(x) cot(x) dx = - csc(x) + C$
15. $\int \frac{1}{\sqrt{a^2 - x^2}} dx = sin^{-1} (\frac{x}{a}) + C$
16. $\int \frac{1}{a^2 + x^2} dx = \frac{1}{a} tan^{-1} (\frac{x}{a}) + C$
17. $\int \frac{1}{x\sqrt{x^2 - a^2}} = \frac{1}{a} sec^{-1} |\frac{x}{a}| + C$

Derivatives

• Derivatives of all six trigonometric functions.
• Derivatives of arcsin(x), arcsec(x) and arctan(x).

Disc/Washer Method

• $\int_{a}^{b} π (R(x)^2 - r(x)^2) dx$
• $\int_{c}^{d} π (R(y)^2 - r(y)^2) dy$

Shell Method

• $\int_{a}^{b} 2π (shell \ radius)(shell \ height) dx$
• $\int_{c}^{d} 2π (shell \ radius)(shell \ height) dy$

Arc Length

• Arc length of a curve $y = f(x)$ between $x = a$ and $x = b$: $L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} dx$

Surface Area

• Area of the surface obtained by revolving the curve $y = f(x)$ about the x-axis between $x = a$ and $x = b$: $S = 2π \int_{a}^{b} f(x)\sqrt{1 + (f'(x))^2} dx$

Standard Trigonometric Identities

• $sin^2(x) + cos^2(x) = 1$
• $1 + tan^2(x) = sec^2(x)$
• $cot^2(x) + 1 = csc^2(x)$
• $cos(2x) = cos^2(x) - sin^2(x)$
• $sin(2x) = 2 sin(x) cos(x)$
• $cos^2(x) = \frac{1 + cos(2x)}{2}$
• $sin^2(x) = \frac{1 - cos(2x)}{2}$

Practice Problems

Section 5.3

• Average Value

  1. Find the average value on the given interval.

     • $f(x) = - \frac{x^2}{2}$ on $[0, 3]$
     • $f(x) = sec(3x) tan(3x)$ on $[0, \frac{π}{12}]$
     • $f(x) = \frac{3}{\sqrt{100 - x^2}}$ on $[0, 5]$
     • $f(x) = cos(5x)\sqrt{sin(5x)}$ on $[\frac{π}{30}, \frac{π}{10}]$

Sections 5.5, 5.6

2. Evaluate the following integrals.

   • $\int x^3e^{x^4} dx$
   • $\int \frac{e^{arcsin(x)}}{\sqrt{1 - x^2}} dx$
   • $\int \frac{1}{x^3\sqrt{1 - \frac{1}{x^2}}} dx$
   • $\int_{e^3}^{e^6} \frac{dt}{tln(t)}$
   • $\int \cos^2(4x) dx$
   • $\int sin^5(4x) cos(4x) dx$
   • $\int \frac{6}{x^2 + 8x + 17} dx$
   • $\int_{0}^{π/9} \frac{sec(3x) tan(3x)}{(5 sec(3x) - 4)^2} dx$
   • $\int \frac{dx}{\sqrt{8x - x^2}}$

3. Find the areas of the regions enclosed by the lines and curves listed below:

   • $y = x^2$ and $y = -x^2 + 4$
   • $y = \frac{1}{x+1}$ and $y = -2x + 1$
   • $x - y^2 = 0$ and $x + 2y^2 = 3$
   • $y = \frac{4}{3+x^2}$ and $y = 1$

4. For the regions sketched below, set-up an expression that calculates the area of the region using (i) a dx-integral, and (ii) a dy-integral. You do not need to evaluate the integrals.

   • The region to the right of the curve $y = x^2 + 1$, above the line $y = 1$ and to the left of the line $y = 9 - 2x$.
   • The region to the right of the y-axis, below the curve $x = 9 - y^2$ and above the line $2x + 9y = 18$.

Sections 6.1, 6.2

5. Find the volumes of solids generated by revolving the regions bounded by the lines and curves below around the x-axis.

   • $y = x^3$, $y = 0$ and $x = 2$
   • $y = 2\sqrt{x}$, $y = 2$ and $x = 0$
   • $x = 2y - y^2$, $x = 0$
   • $y = \sqrt{x}$, $y = 0$, $y = 2 - x$

6. Find the volumes of solids generated by revolving the regions bounded by the lines and curves below around the y-axis.

   • $y = x$, $y = -x/2$, $x = 2$
   • $y = 2x - 1$, $y = \sqrt{x}$, $x = 0$
   • $x = y^{3/2}$, $x = 0$, $y = 2$
   • $x = \sqrt{2y+1}$, $x = 0$, $y = 0$, $y = 3$

7. Find the volume obtained by revolving each region described below about the indicated axis.

   • The region bounded by the curves $y = x^2 + 1$ and $y = x + 3$ revolved about the line $y = 1$.
   • The region to the right of the y-axis bounded by $y = x^2$ and $y = 2 - x$ revolved about the line $x = -3$.
   • The region bounded by $y = e^{-x}$, $y = 0$, $x = 0$ and $x = 1$ revolved about the line $y = 2$.
   • The region bounded by the lines $2y = x + 4$, $y = x$ and $x = 0$ revolved about the line $x = 4$.

8. Consider the region R bounded by the curve $y = ln(x - 1)$, the line $x = 2$ and the line $y = 3$.

   • Set-up integrals computing the volume of the solid obtained by revolving R about each axis given below using (i) the disk/washer method, and (ii) the shell method. Do not evaluate the integrals.
     • $y = 3$
     • y-axis
     • $y = -4$
     • $x = 1$

9. The triangular region bounded by the line $y = 6-2x$, the x-axis and the y-axis is the base of a solid. Find the volume of the solid if

   • the cross-sections perpendicular to the x-axis are semi-circles with diameter in the base.
   • the cross-sections perpendicular to the y-axis are squares.

Sections 6.3, 6.4

10. Find the arc lengths of the following curves on the given intervals:

    • $f(x) = x^{3/2}$ on $0 \le x \le 4$
    • $f(y) = \frac{y^{3/2}}{3} - y^{1/2}$ on $1 \le y \le 9$
    • $f(x) = \frac{x^4}{4} + \frac{1}{8x^2}$ on $1 \le x \le 2$
    • $f(x) = \frac{e^x+e^{-x}}{2}$ on $-1 \le x \le 1$
    • $f(x) = \frac{1}{3} ln(|cos(3x)|)$ on $\frac{π}{4} \leqslant x \leqslant \frac{π}{3}$
    • $f(x) = \sqrt{36 - x^2}$ for $0 \leqslant x \leqslant 3$.

11. Find the areas of the surface generated when the given curves are revolved around the given axis.

    • $y = 3x + 4$ for $0 \le x \le 6$ around the x-axis.
    • $y = 8\sqrt{x}$ for $9 \le x \le 20$ around the x-axis.
    • $y = \frac{x^2}{4}$ for $2 \le x \le 4$ around the y-axis.
    • $x = \sqrt{16y - y^2}$ for $0 \leqslant y \leqslant 8$ revolved about the y-axis.

12. Let C be the curve $y = ln(2x)$ on the interval $2 \le x \le 7$. Set up (but do not evaluate) an integral to find the area of the surface generated when C is revolved about the specified axis.

    • Axis: $y = -2$ (set up as a dx integral)
    • Axis: $y = -2$ (set up as a dy integral)
    • Axis: $x = 10$ (set up as a dx integral)
    • Axis: $x = 10$ (set up as a dy integral)

Section 8.2

13. Evaluate the following integrals:

    • $\int x^2 cos(x) dx$
    • $\int x sec^2(x) dx$
    • $\int \frac{ln x}{x^2} dx$
    • $\int x sin(2x) dx$
    • $\int arcsin(x) dx$
    • $\int e^{2x} cos(x) dx$
    • $\int x^2e^{-x} dx$
    • $\int \frac{x}{\sqrt{x - 1}} dx$
    • $\int \sqrt{x} ln(x) dx$

More Practice Problems

14. Evaluate the following integrals.

    • $\int \frac{1}{(2x + 1)\sqrt{4x + 4x^2}} dx$
    • $\int \frac{2x + 7}{\sqrt{x + 1}} dx$
    • $\int \frac{1}{e^x + e^{-x}} dx$
    • $\int \frac{1}{\sqrt{-x^2 + 6x}} dx$
    • $\int \frac{1}{\sqrt{e^{2x} - 1}} dx$
    • $\int_{-4}^{2} \sqrt{8 - 2x - x^2} dx$
    • $\int \frac{xe^x}{(x + 1)^2} dx$
    • $\int arctan(x) dx$
    • $\int e^{8x} sin(3x) dx$
    • $\int e^{\sqrt{5x+3}} dx$
    • $\int x^2 arctan(x) dx$

15. Find the average value of $f(x) = x^7 + x^2 + sin x + |2x|\sqrt{x^2 + 1}$ on the interval $[-3, 3]$.

16. Let $F(x)$ be an anti-derivative of $f(x) = \frac{sin(x)}{x}$ for x > 0. Express $\int_{1}^{3} \frac{sin(2x)}{x} dx$ in terms of $F(x)$.

17. Find the area of the ‘triangular’ region in the first quadrant that is bounded above by the curve $y = e^{x/2}$, below by $y = e^{-x/2}$ and on the right by $x = 2 ln(2)$.

18. Consider the region R inside the circle $x^2 + y^2 = 25$ and above the horizontal line $y = 3$.

    • Set up an x-integral that calculates the area of the region R. Do not evaluate the integral.
    • Set up a y-integral that calculates the area of the region R. Do not evaluate the integral.

19. Let R be the region bounded by $x = y - y^3$ and the y-axis. Find the volume of the solid generated by revolving R around the line $y = 1$.

20. Let R be the region bounded by $y = 2x - x^2$ and $y = x$. Find the volume of the solid generated by revolving R around the line $x = 1$.

21. The region R is bounded by $y = e^{x/2}$, $y = 7$ and $x = 2$. We create a solid of revolution by revolving R about the line $x = -3$.

    • Set up an integral that computes the volume of the solid using the method of cylindrical shells. Do not evaluate the integral.
    • Set up an integral that computes the volume of the solid using the disk/washer method. Do not evaluate the integral.

22. The region R in the plane bounded by $y = \sqrt{x}$ and $y = x/2$ is the base of a solid. Find the volume of the solid if:

    • The cross-sections perpendicular to the x-axis are squares.
    • The cross-sections perpendicular to the x-axis are isosceles triangles of height 6.
    • The cross-sections perpendicular to the y-axis are circular disks whose diameters traverse R.

23. A region R is bounded in the plane by the lines: $y = -x + 3$, $y = 2x - 6$ and $y = 3$. Find the volume of the solid given by rotating R around the line $x = 5$.

24. Set up an integral that computes the perimeter of the ellipse $\frac{x^2}{4} + \frac{y^2}{25} = 1$.

25. Find the length of the curve $y = \int_{0}^{x} cos(2t) dt$ from $x = 0$ to $x = \frac{π}{4}$.

26. Let L be the line segment between $(1, 2)$ and $(4, -4)$. Find the area of the surface generated when L is revolved around the line $y = 2$.

27. Let L be the line segment between $(1, 2)$ and $(4, -4)$. Find the area of the surface generated when L is revolved around the line $x = -1$.

28. Set-up integrals to compute the surface area of the cone whose base has radius 4 and has height 5.

29. Let R be the region under the graph of $y = \sqrt{\frac{1}{16-x^2}}$ for $0 \leqslant x \leqslant 2$. Find the volume of the solid obtained by revolving R about the line $x = -3$.

30. Consider the region R between the graphs of $y = sec(x)$ and $y = tan(x)$ for $0 \leqslant x \leqslant \frac{π}{4}$.

    • Find the area of the region.
    • Find the volume of the solid obtained by revolving the region about the x-axis.
    • Find the volume of the solid whose base is the region R and whose cross-sections perpendicular to the x-axis are squares.

31. Find the arc length of the graph of $f(x) = \frac{4}{5}x^{5/4}$ for $0 \leqslant x \leqslant 9$.

32. Use integration by parts to derive the reduction formula for $\int sin^n x dx$:

    • $\int sin^n x dx = - \frac{1}{n} sin^{n-1} x cos x + \frac{n - 1}{n} \int sin^{n-2} x dx$

33. Use integration by parts to rewrite $\int csc^7(3x) dx$ in terms of $\int csc^5(3x) dx$.