Can anyone help with the following problem please...Thank u so much
Let f(x) =|9x|. For the area above the x-axis and between the lines x= -9 and x=18
a. draw and label a sketch with
the area shaded
b. find the area of the region in
the xy-plane under the graph
of f(x)
For this kind of problem, it is always desirable to make a sketch within the limits of the domain, from x=-9 to 18.
See:
http://img510.imageshack.us/img510/2928/1291868155.png
Using the formula for the area of a triangle as:
Area = (1/2)base*height
You can find the areas of the two triangles between the function and the x-axis.
If you need further help, don't hesitate to post.
Amy, please check to see if your question has been answered before posting the same question again.
you asked the same question with only a small change yesterday,
http://www.jiskha.com/display.cgi?id=1291781963
MathMate has gone to some length to answer it again, even creating a picture for you.
make a note of your post by jotting down the time of the first posting, even though it moves off the page, it is then rather easy to find
Thank you for your help
To solve this problem, we will follow these steps:
a. To draw and label a sketch with the shaded area, we need to consider the function f(x) = |9x| for the given range -9 ≤ x ≤ 18.
Start by drawing the x-axis and marking the points -9 and 18 on the axis. Then, consider the function f(x) = |9x|.
For values of x between -9 and 0, the function takes the form f(x) = -9x. Draw a line passing through the origin (0,0) with a negative slope of -9.
For values of x between 0 and 18, the function takes the form f(x) = 9x. Draw a line passing through the origin (0,0) with a positive slope of 9.
Now, shade the region above the x-axis and between the lines x = -9 and x = 18. The shaded region represents the area we are interested in.
b. To find the area of the region in the xy-plane under the graph of f(x), we need to calculate the integral of f(x) over the given range (-9 to 18). Remember that when calculating the integral of a function, we are finding the signed area under the curve.
For this problem, we need to divide the given range into two parts: -9 to 0 and 0 to 18.
The integral of f(x) = |9x| from -9 to 0 is given by:
∫f(x)dx = ∫-9x dx = -4.5x^2 | -9 to 0
= -4.5(0)^2 - (-4.5(-9)^2)
= -4.5(0 - 81)
= -4.5(-81)
= 364.5
The integral of f(x) = |9x| from 0 to 18 is given by:
∫f(x)dx = ∫9x dx = 4.5x^2 | 0 to 18
= 4.5(18)^2 - 4.5(0)^2
= 4.5(324) - 0
= 1458
Now, add the results of both integrals to find the total area under the graph of f(x):
Total Area = 364.5 + 1458
= 1822.5
Therefore, the area of the region in the xy-plane under the graph of f(x) is 1822.5 square units.