Area of a bounded region using integration

[Steven_Scott]

In Calculus II, we're currently learning how to find the area of a bounded region using integration. My professor wants us to solve a problem where we're given a graph of two arbitrary functions, f(x) and g(x) and their intersection points, labeled (a,b) and (c,d) with nothing else given.

I know how to find set up the integration if I'm given the functions explicitly but these aren't specified. Everything I find in textbooks and on-line only gives examples with specified functions, not arbitrary ones labeled f(x) and g(x).

Can someone please show me how I'd go about setting up a problem like this?

 

[PeroK]

In Calculus II, we're currently learning how to find the area of a bounded region using integration. My professor wants us to solve a problem where we're given a graph of two arbitrary functions, f(x) and g(x) and their intersection points, labeled (a,b) and (b,c) with nothing else given.

I know how to find set up the integration if I'm given the functions explicitly but these aren't specified. Everything I find in textbooks and on-line only gives examples with specified functions, not arbitrary ones labeled f(x) and g(x).

Can someone please show me how I'd go about setting up a problem like this?

If $f(x) = x^2$ and $g(x) = x$ what would you do and why?

You then need to generalise this approach.

 

[kuruman]

... where we're given a graph of two arbitrary functions, f(x) and g(x) and their intersection points, labeled (a,b) and (b,c) ...

Is there a typo here? Shouldn't the intersection points be (a,b) and (c,d)?

 

[Steven_Scott]

Is there a typo here? Shouldn't the intersection points be (a,b) and (c,d)?

You're right.
Thank you.

 

[epenguin]

I don't understand how you can possibly be given a formulation that is only for particular functions and not general. E.g for the area delimited by the x-axis, a function f(x) and x = a and x = b for example.
The answer to this question will not be the most neat compressed formula, but one of those more strung-out 'peicewise' mones involving x≥, x≤ etc. that you have surely met and done exercises on, e.g. when they were explaining what what is meant by a function.

 

[kuruman]

$f(x)dx$ is an elemental area "under the curve" consisting of a rectangle of height $f(x)$ and width $dx$ located at some $x$.
$g(x)dx$ is an elemental area"under the curve" consisting of a rectangle of height $g(x)$ and width $dx$ located at the same $x$.
$dA=f(x)dx-g(x)dx$ is the "area under the curve" difference between $f(x)dx$ and $g(x)dx$ also at the same $x$.
What is the total "area under the curve" difference in the interval $x_1 \le x \le x_2$?
Hint Add all the $dA$'s together and write an expression for the sum in terms of the given symbols.