Draw a diagram to interpret this equation geometrially as an equality

[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:

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

$$u(1)=0$$

$$u(0)=1$$

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

 

[arildno]

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

 

[DivGradCurl]

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

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.

 

[arildno]

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

 

[DivGradCurl]

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

I've got it. Thanks!