# Draw a diagram to interpret this equation geometrially as an equality

In summary, we discussed three problems involving integrals and functions. For the first problem, we proved the equality of areas for a continuous function on the real numbers. For the second problem, we used the substitution rule to show that the equality of areas holds for a translated function. Finally, for the third problem, we applied the substitution rule again to show that the equality of areas still holds for positive numbers a and b. Additionally, we discussed how to interpret these equations geometrically using diagrams.
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:
This looks fine to me.
Where do you need help?

arildno said:
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.

Last edited:
For diagram 2, remember that f(x+c) represents a translation.

arildno said:
For diagram 2, remember that f(x+c) represents a translation.

I've got it. Thanks!

## 1. What does it mean to "draw a diagram to interpret an equation geometrially"?

Interpreting an equation geometrically means representing the equation in a visual form, such as a graph or diagram. This helps to better understand the relationship between the variables in the equation.

## 2. How can a diagram help in understanding an equation?

A diagram can provide a visual representation of the equation, making it easier to visualize the relationship between the variables and understand how they interact with each other.

## 3. What does it mean to interpret an equation as an equality?

An equation is a statement that shows the equality between two expressions. Interpreting an equation as an equality means understanding that the two sides of the equation are equal to each other.

## 4. What type of equations can be interpreted geometrically?

Most equations can be interpreted geometrically, as long as they have two or more variables that can be represented on a graph or diagram.

## 5. How can drawing a diagram help in solving an equation?

Drawing a diagram can help in solving an equation as it provides a visual representation of the equation, making it easier to understand and manipulate. It can also help in identifying patterns and relationships between the variables, which can aid in finding a solution.

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