By Mark Zegarelli

**An easy-to-understand primer on complicated calculus topics**

Calculus II is a prerequisite for plenty of well known university majors, together with pre-med, engineering, and physics. *Calculus II For Dummies* deals professional guideline, suggestion, and how you can aid moment semester calculus scholars get a deal with at the topic and ace their tests. It covers intermediate calculus issues in simple English, that includes in-depth assurance of integration, together with substitution, integration thoughts and whilst to exploit them, approximate integration, and fallacious integrals. This hands-on advisor additionally covers sequences and sequence, with introductions to multivariable calculus, differential equations, and numerical research. better of all, it comprises useful workouts designed to simplify and increase figuring out of this advanced topic.

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**Additional resources for Calculus II For Dummies**

**Example text**

You might imagine that by slicing the area into more rectangles (say 10, or 100, or 1,000,000), you’d get progressively better estimates. 3333.... 33 1 This, in fact, is the correct answer. But to justify this conclusion, you need a bit more rigor. Building a formula for finding area In the previous section, you calculate the areas of two rectangles and four rectangles, respectively, as follows: 2 (1) + 2 (9) = 2 (1 + 9) = 20 1 (1) + 1 (4) + 1 (9) + 1 (16) = 1 (1 + 4 + 9 + 16) = 30 Each time, you divide the area that you’re trying to measure into rectangles that all have the same width.

For example, here’s an integral that can’t be evaluated: #e x2 dx But even though you can’t solve this integral, you can approximate its solution to any degree of accuracy that you desire. And for real-world applications, a good approximation is often acceptable as long as you (or, more likely, a computer) can calculate it in a reasonable amount of time. Such a procedure for approximating the solution to a problem is called an algorithm. Numerical analysis examines algorithms for qualities such as precision (the margin of error for an approximation) and tractability (how long the calculation takes for a particular level of precision).

X x=1 x=5 In this case, the upper-right corner touches the function, so the heights of the four rectangles are f(2), f(3), f(4), and f(5). Now, suppose that I draw the rectangles as shown in Figure 1-14. y y = x2 Figure 1-14: Approximating area with midpoint rectangles. 5). Chapter 1: An Aerial View of the Area Problem It seems that I can draw rectangles at least three different ways to approximate the area that I’m attempting to measure. They all lead to different approximations, so which one leads to the correct answer?