can not figure out problem.
(the area of the region in the first quadrant between the graph of y=x sqrt (4-x^2) and the axis is ?
a(2/3)sqrt2
b 8/3
c 2sqrt2
d 2sqrt3
e 16/3
I agree with Mgraph.
To find the area of the region in the first quadrant between the graph of y = x sqrt(4 - x^2) and the x-axis, you can use integration.
First, let's find the x-intercepts of the graph:
Setting y = 0:
0 = x sqrt(4 - x^2)
Either x = 0 or 4 - x^2 = 0
If 4 - x^2 = 0, then x^2 = 4, and x = ±2
So the graph intersects the x-axis at x = 0 and x = 2.
To find the area between the x-axis and the graph, integrate the equation from x = 0 to x = 2:
∫[0 to 2] x sqrt(4 - x^2) dx
To simplify the integration, use the substitution u = 4 - x^2, du = -2x dx:
∫[0 to 2] -1/2 sqrt(u) du
= -1/2 ∫[0 to 2] u^(1/2) du
= -1/2 * 2/3 * u^(3/2) |[0 to 2]
= -1/3 * (4 - 0)
= -4/3
Since we are looking for the area, take the absolute value:
| -4/3 | = 4/3
So, the area of the region in the first quadrant is 4/3.
Unfortunately, none of the given options match the correct answer.
To find the area of the region in the first quadrant between the graph of y = x * sqrt(4 - x^2) and the x-axis, you can use integration. Here's the step-by-step process:
1. First, find the x-intercepts of the given function by setting y = 0:
0 = x * sqrt(4 - x^2)
This equation is satisfied when x = 0 (at the origin) and when 4 - x^2 = 0 (giving x = ±2).
2. Next, determine the boundaries for integration. Since you are interested in the area between the function and the x-axis in the first quadrant, the integration limits should be from x = 0 to x = 2.
3. Set up the integral for finding the area. Since the function is y = x * sqrt(4 - x^2), the area can be calculated using a definite integral:
Area = ∫[0 to 2] (x * sqrt(4 - x^2)) dx
4. Simplify and evaluate the integral:
∫(x * sqrt(4 - x^2)) dx = -√(4 - x^2) + C
5. To find the definite integral, substitute the upper and lower limits into the integral and evaluate:
Area = -√(4 - 2^2) - (-√(4 - 0^2))
= -√0 - (-√4)
= -0 - (-2)
= 2
Therefore, the area of the region in the first quadrant between the graph of y = x * sqrt(4 - x^2) and the x-axis is 2.
Among the given options:
a) (2/3)√2
b) 8/3
c) 2√2
d) 2√3
e) 16/3
The correct answer is b) 8/3.
f(x)>0 on(0,2) and f(0)=f(2)=0
The primitive of f(x) is
-(4-x^2)sqrt(4-x^2)/3
The area=-(4-2^2)sqrt(4-2^2)/3+(4-0^2)sqrt(4-0^)/3=8/3