Chapter 4: Q. 58 (page 353)

Give a geometric argument to prove Theorem 4.13(a): For any real numbers a, b, and c,
$\int_{a}^{b} c d x = c (b - a)$
(Hint: Use a rectangle.)

Short Answer

Expert verified

The theorem 4.13(a) is proved.

$\int_{a}^{b} c d x = c (b - a)$

Step by step solution

Step 1. Given Information

We are given a theorem,

$\int_{a}^{b} c d x = c (b - a)$

Step 2. Proving the theorem

Take, $f (x) = c$ on $[a, b]$ . Then the integral $\int_{a}^{b} f (x) d x$ , represents the area from a to b under the constant curve $f (x) = c$ . That is, the integral $\int_{a}^{b} f (x) d x$ , represents the area of the rectangle of length $(b - a)$ and breadth c.

Hence Proved.

$\int_{a}^{b} f (x) d x = c (b - a)$ .

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Give a geometric argument to prove Theorem 4.13(a): For any real numbers a, b, and c,
$\int_{a}^{b} c d x = c (b - a)$
(Hint: Use a rectangle.)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the theorem

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Give a geometric argument to prove Theorem 4.13(a): For any real numbers a, b, and c,∫abcdx=c(b-a)(Hint: Use a rectangle.)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the theorem

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Discrete Mathematics

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Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Give a geometric argument to prove Theorem 4.13(a): For any real numbers a, b, and c,
$\int_{a}^{b} c d x = c (b - a)$
(Hint: Use a rectangle.)