Seeking step by step help with this question...Thank you much.
Let f(x) =|9x|. For the area above the x-axis and between the lines x= -9 and x=18
I made a sketch, and y = |9x| consists of two lines
first: y = 9x starting at the origin, and the second:
y = -9x also starting in the origin,
resulting in a graph looking like a V
If I understand the question correctly, you now want the area of the triangle in the second quadrant and the triangle in the first quadrant
left triangle: base = 8 , height = 72 , Area = 8*72/2=288
right triangle: base = 18, height = 162, Area = 1456
area = 288+1456 = 1746
To find the area above the x-axis and between the lines x = -9 and x = 18 for the function f(x) = |9x|, you can use the definite integral. Here are the step-by-step instructions:
Step 1: Identify the limits of integration. In this case, the limits are x = -9 and x = 18.
Step 2: Determine the integrand. Since we are calculating the area above the x-axis, we want to integrate the positive portion of the function. The function |9x| splits into two linear functions: 9x when x >= 0 and -9x when x < 0. However, for the range x = -9 to x = 0, we need to consider that |9x| = -9x. So, the integrand becomes:
f(x) = 9x when 0 <= x <= 18 and f(x) = -9x when -9 <= x < 0.
Step 3: Set up the integral. The integral will be a combination of two integrals since the function changes at x = 0. Split the integral at x = 0:
∫[from -9 to 0] -9x dx + ∫[from 0 to 18] 9x dx.
Step 4: Evaluate the integrals. To evaluate the integral of each portion separately, you need to apply the power rule of integration. The integral -9x dx equals (-9/2)x^2, and the integral 9x dx equals (9/2)x^2.
Evaluating the first portion:
∫[from -9 to 0] -9x dx = (-9/2) [(0)^2 - (-9)^2] = (-9/2) (0 - 81) = 364.5.
Evaluating the second portion:
∫[from 0 to 18] 9x dx = (9/2) [(18)^2 - (0)^2] = (9/2) (324 - 0) = 1458.
Step 5: Combine the results. Since we are finding the area, we need to take the absolute value of each integral result and sum them together:
|364.5| + 1458 = 1822.5.
Therefore, the area above the x-axis and between the lines x = -9 and x = 18 for the function f(x) = |9x| is 1822.5 square units.