Cracking The Ap Calculus ab Exam 2018

If you slice the region vertically and revolve the slice, you won’t get a washer; you’ll get a cylinder instead. ...

From a general standpoint: If we have a region whose area is bounded above by the curve y = f(x) ...

Sketch the figure. If you slice the region vertically, the height of the shell doesn’t change because of the shift ...

Let your sketch be your guide. Each horizontal slice is bounded on the right by the curve x = y and o ...

The radius of each cylinder is increased by 1 because of the shift in the axis of revolution, so the inte ...

What this problem is telling us is that every time we make a vertical slice, the slice is the leng ...

As you can see, the technique is very simple. First, you find the side of the cross-section in terms of ...

. Now because the area of a semi-circle is (diameter)^2 , we can find the volume by evaluating the integral ...

looks like the following: π [(16 − x^2 )^2 − (16 − 4x)^2 ] dx PROBLEM 2. Repeat Problem 1, but revolve the re ...

To slice horizontally, you have to solve each equation for x in terms of y and find the limits of ...

Slicing horizontally, the curve x = is always on the right and the curve x = is always on the left. ...

If you were to slice the region vertically, you would use washers. You’ll need to add 3 to each radius ...

If you choose cylindrical shells, slice the region vertically; you’ll need to adjust for the axis of revolution. ...

Expand the integrand to get (16 − 8x^2 + x^4 ) dx Then integrate, which gives you PRACTICE PROBLEM SET 2 ...

8.Use the method of cylindrical shells to find the volume of the solid that results when the region bounded b ...

Chapter 18 Differential Equations ...

There are many types of differential equations, but only a very small number of them appear on the ...

∫ = ∫^3 x (^2) dx The result is ln y = x^3 + C. It’s customary to solve this equation for y. Y ...

Integrate the expression: h = −16t^2 + 64t + C. Now solve for the constant by plugging in t = 0 and ...

Answer: First, separate the variables: = 4x dx. Then, take the integral of both sides. ∫ = ∫^4 x ...