Tony Hartman

Massachusetts Institute of Technology
Professor

Biography

I have taught freshman and sophomore mathematics for 40 years.

Education

BS Physics
Massachusetts Institute of Technology

Educator Statistics

Numerade tutor for 4 years
272 Students Helped

Topics Covered

Differential Equations Made Simple: Expert Tips & Resources
Applications of Integration: Exploring Real-World Solutions
Mastering Integration Techniques for Optimal Results

Tony's Textbook Answer Videos

02:08
Calculus Early Transcendentals

Assume $f$ and $g$ are continuous with $f(x) \geq g(x)$ on $[a, b] .$ The region bounded by the graphs of $f$ and $g$ and the lines $x=a$ and $x=b$ is revolved about the $y$ -axis. Write the integral given by the shell method that equals the volume of the resulting solid.

Chapter 6: Applications of Integration
Section 4: Volume by Shells
Tony Hartman
01:53
Calculus Early Transcendentals

Let $R$ be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when $R$ is revolved about the $y$ -axis.
$$y=\left(1+x^{2}\right)^{-1}, y=0, x=0, \text { and } x=2$$

Chapter 6: Applications of Integration
Section 4: Volume by Shells
Tony Hartman
02:02
Calculus Early Transcendentals

Let $R$ be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when $R$ is revolved about the $y$ -axis.
$y=3 x, y=3,$ and $x=0$ (Use integration and check your answer using the volume formula for a cone.)

Chapter 6: Applications of Integration
Section 4: Volume by Shells
Tony Hartman
01:23
Calculus Early Transcendentals

Let $R$ be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when $R$ is revolved about the $y$ -axis.
$$y=\sqrt{x}, y=0, \text { and } x=1$$

Chapter 6: Applications of Integration
Section 4: Volume by Shells
Tony Hartman
01:51
Calculus Early Transcendentals

Let $R$ be the region bounded by $y=x^{2}, x=1,$ and $y=0 .$ Use the shell method to find the volume of the solid generated when $R$ is revolved about the following lines.
$$x=1$$

Chapter 6: Applications of Integration
Section 4: Volume by Shells
Tony Hartman
1 2 3 4 5 ... 9

Tony's Quick Ask Videos

02:18
Calculus 1 / AB

(12 pts) A farmer wishes to make a small rectangular garden with
one side against the barn. If he has 200 feet of fence, find the
dimensions of the garden of maximum area.

Tony Hartman
02:25
Calculus 3

The sides of a rectangular box are measured to have lengths 5, 6, and 8 (centimeters). The error in measuring each side is at most 0.1 cm. Use differentials to estimate the maximal error in measuring the volume of the box.

Tony Hartman
03:36
Calculus 3

Solve the given DE using Laplace transform:
(dx/dt) + 9x = 12, where x(0) = 1.

Tony Hartman
04:21
Calculus 3

The ends of a "parabolic" water tank are the shape of the region
inside the graph
of y = x2 for 0 ≤ y ≤ 9 ;
the cross sections parallel to the top of the tank (and the ground)
are rectangles. At its center the tank is 9 feet deep and 6 feet
across. The tank is 6 feet long. Rain has filled the tank and water
is removed by pumping it up to a spout that is 4 feet above the top
of the tank. Set up a definite integral to find the work W that is
done to lower the water to a depth of 5 feet and then find the
work.

Tony Hartman
05:12
Calculus 1 / AB

Find the center of mass of a thin plate of constant
density δ covering the given region.
The region bounded by the parabolas y=2x^2-4x and y=2x-x^2

Tony Hartman
03:46
Calculus 1 / AB

Find the area of the region that lies inside the first curve and
outside the second curve.
r =
14 cos θ, r =
7

Tony Hartman
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