If m ≤ f(x) ≤ M for a ≤ x ≤ b, where m is the absolute minimum and M is the absolute maximum of f on the interval [a, b], then
b
m(b − a) ≤ f(x) dx ≤ M(b − a).
a
Use this property to estimate the value of the integral.
9
4Sqrt(x)dx
0
Correct: Your answer is correct.
(smaller value)
Incorrect: Your answer is incorrect.
(larger value)
just do a direct substitution
0 <= 4√x <= 4√9
0 <= 4√x <= 12 on [0,9]
so,
0*9 <= ∫4√x <= 12*9
To estimate the value of the integral \(\int_{0}^{9} 4\sqrt{x} dx\), we can use the given property.
Let's first find the absolute minimum and maximum of \(f(x) = 4\sqrt{x}\) on the interval \([0, 9]\).
To find the absolute minimum, we can set \(f'(x) = 0\):
\(\frac{df}{dx} = \frac{d}{dx} (4\sqrt{x}) = \frac{4}{2\sqrt{x}} = \frac{2}{\sqrt{x}}\).
Setting \(\frac{2}{\sqrt{x}} = 0\), we get \(x = 0\).
Therefore, the absolute minimum occurs at \(x = 0\) and the corresponding value is \(m = f(0) = 4\sqrt{0} = 0\).
To find the absolute maximum, we can examine the endpoints as well as the critical points. However, since \(f(x)\) is an increasing function, the absolute maximum will occur at the endpoint \(x = 9\).
Therefore, the absolute maximum occurs at \(x = 9\) and the corresponding value is \(M = f(9) = 4\sqrt{9} = 12\).
Now, we can use the given property to estimate the value of the integral:
\(m(b-a) \leq \int_{0}^{9} 4\sqrt{x} dx \leq M(b-a)\).
Substituting the values we found, we get:
\(0(9-0) \leq \int_{0}^{9} 4\sqrt{x} dx \leq 12(9-0)\).
Simplifying, we have:
\(0 \leq \int_{0}^{9} 4\sqrt{x} dx \leq 108\).
Therefore, the value of the integral \(\int_{0}^{9} 4\sqrt{x} dx\) is estimated to be between 0 and 108.
To estimate the value of the integral ∫ 4√(x) dx from 0 to 9 using the given property, we need to find the absolute minimum and maximum of the function 4√(x) on the interval [0, 9].
First, let's find the derivative of the function f(x) = 4√(x):
f'(x) = 2(1/2)(x)^(-1/2) = 2/√(x)
Next, we can set the derivative equal to zero to find any critical points:
2/√(x) = 0
This equation has no real solutions, so there are no critical points within the interval [0, 9].
Now, let's check the endpoints of the interval:
f(0) = 4√(0) = 0
f(9) = 4√(9) = 4(3) = 12
Therefore, the absolute minimum of f(x) on the interval [0, 9] is 0 and the absolute maximum is 12.
Using the property, we can write the following inequality:
0(9-0) ≤ ∫ 4√(x) dx ≤ 12(9-0)
Simplifying the inequality, we get:
0 ≤ ∫ 4√(x) dx ≤ 108
Therefore, the estimated value of the integral is between 0 and 108.
Since the correct answer is given as "smaller value," we can conclude that the estimated value of the integral is 0.