Homework Help: Draw a diagram to interpret this equation geometrially as an equality

1. Apr 23, 2005

DivGradCurl

I need some help with the following problems. Any help is highly appreciated.

1. If $$f$$ is continuous on $$\mathbb{R}$$, prove that

$$\int _a ^b f(-x) \: dx = \int _{-b} ^{-a} f(x) \: dx$$

For the case where $$f(x) \geq 0$$ and $$0 < a < b$$, draw a diagram to interpret this equation geometrially as an equality of areas.

2. If $$f$$ is continuous on $$\mathbb{R}$$, prove that

$$\int _a ^b f(x + c) \: dx = \int _{a+c} ^{b+c} f(x) \: dx$$

For the case where $$f(x) \geq 0$$, draw a diagram to interpret this equation geometrially as an equality of areas.

3. If $$a$$ and $$b$$ are positive numbers, show that

$$\int _0 ^1 x^a (1 - x) ^b \: dx = \int _0 ^1 x^b (1 - x) ^a \: dx$$

Here is what I've got so far:

1. Consider the left-hand side

$$\int _a ^b f(-x) \: dx$$

and apply the substitution rule:

$$u=-x \Rightarrow \frac{du}{dx} = -1 \Rightarrow dx = - du$$

$$u(b)=-b$$

$$u(a)=-a$$

$$\int _a ^b f(-x) \: dx = -\int _{-a} ^{-b} f(u) \: du = \int _{-b} ^{-a} f(u) \: du = \int _{-b} ^{-a} f(x) \: dx$$

2. Consider the left-hand side

$$\int _a ^b f(x + c) \: dx$$

and apply the substitution rule:

$$u=x+c \Rightarrow \frac{du}{dx} = 1 \Rightarrow dx = du$$

$$u(b)=b+c$$

$$u(a)=a+c$$

$$\int _a ^b f(x + c) \: dx = \int _{a+c} ^{b+c} f(u) \: du = \int _{a+c} ^{b+c} f(x) \: dx$$

3. Consider the left-hand side

$$\int _0 ^1 x^a (1 - x) ^b \: dx$$

and apply the substitution rule:

$$u=1-x \Rightarrow \frac{du}{dx} = -1 \Rightarrow dx = -du$$

$$u(1)=0$$

$$u(0)=1$$

$$\int _0 ^1 x^a (1 - x) ^b \: dx = \int _1 ^0 u^b (1 - u) ^a \: du = \int _0 ^1 x^b (1 - x) ^a \: dx$$

Last edited: Apr 23, 2005
2. Apr 23, 2005

arildno

This looks fine to me.
Where do you need help?

3. Apr 23, 2005

DivGradCurl

Oh, good! Thanks for checking it out.

Other than that, I need to take care of the diagrams (problems 1 & 2). I'm not so sure how to handle those. It seems that the areas in #1 are symmetric with respect to the y-axis.

Last edited: Apr 23, 2005
4. Apr 23, 2005

arildno

You're right about diagram 1.
For diagram 2, remember that f(x+c) represents a translation.

5. Apr 23, 2005

DivGradCurl

I've got it. Thanks!

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook