∫ ( x² - 4 x + 4 ) dx = x³ / 3 - 4 x² / 2 + 4 x + C
∫ ( x² - 4 x + 4 ) dx = x³ / 3 - 2 x² + 4 x + C
The area under the graph of a continuous function f(x) between the vertical lines x = a and x = b
can be computed by the definite integral:
b
∫ f(x) dx = F(b) - F(a)
a
In this case:
F(a) = F(0) = 0³ / 3 - 2 ∙ 0² + 4 ∙ 0 = 0
F(b) = F(2) = 2³ / 3 - 2 ∙ 2² + 4 ∙ 2 = 8 / 3 - 2 ∙ 4 + 4 ∙ 2 = 8 / 3 - 8 + 8 = 8 / 3
2
∫ ( x² - 4 x + 4 ) dx = F(2) - F(0) = 8 / 3 - 0 = 8 / 3
0
Now you must find the definite integral such as to be:
h
∫ ( x² - 4 x + 4 ) dx = F(h) - F(0) = 1 / 2 ( 8 / 3 ) = 8 / 6 = 2 ∙ 4 / 2 ∙ 3 = 4 / 3
0
F(h) = h³ / 3 - 2 h² + 4 h
F(0) = 0
F(h) - F(0) = F(h)
This means that you need to calculate:
F(h) = 4 / 3
h³ / 3 - 2 h² + 4 h = 4 / 3
Multiply both sides by 3
h³ - 6 h² + 12 h = 4
Now you must solve:
h³ - 6 h² + 12 h - 4 = 0
This equation can only be solved by numerical methods.
Using Newton-Raphson method:
x ≈ 0.41259
The region R is the region in the first quadrant bounded by the curves y=x^ 2 -4x+ 4, x = 0 , and x = 2, as seen in the image attached below:
ibb.co/WgrRRRL
Find a value h such that the vertical line x = h divides the region R into two regions of equal area. You get h = ?
I think the answer is 0.39987, but I could be wrong.
Help is greatly appreciated!
2 answers
Thank you.