Find the area of the region under the graph of the function f on the interval [4, 10].
f(x)=10x-5
? square units
"the area of the region under the graph" is an integral of f(x) from a to b, in your case from 4 to 10.
∫ f(x) from 4 to 10=∫ (10x-5)dx from 4 to 10.
you just have to compute the integral and that will give you:
1/2*10x^2-5x evaluate it from 4 to 10
I hope it helps
thanx! :) i got it!
To find the area of the region under the graph of the function f on the interval [4, 10], we can use the concept of definite integration. The area under the curve is given by the definite integral of the function over the given interval.
The definite integral of a function f(x) from a to b is denoted as ∫f(x) dx, where a and b represent the limits of integration.
In this case, we want to find the area under the curve of the function f(x) = 10x - 5 over the interval [4, 10]. The area is given by:
Area = ∫[4, 10] (10x - 5) dx
We can integrate the function term by term. The integral of 10x with respect to x is (10/2)x^2, and the integral of -5 with respect to x is -5x.
Applying the Fundamental Theorem of Calculus, the definite integral of f(x) from a to b is F(b) - F(a), where F(x) is the antiderivative of f(x).
Applying the antiderivative to each term of the function, we have:
Area = [(10/2)x^2 - 5x] from 4 to 10
Evaluating the integral at the limits of integration, we find:
Area = [(10/2)(10^2) - 5(10)] - [(10/2)(4^2) - 5(4)]
Area = [500 - 50] - [80 - 20]
Area = 450 - 60
Area = 390 square units
Therefore, the area of the region under the graph of the function f on the interval [4, 10] is 390 square units.
To find the area of the region under the graph of the function f(x) on the interval [4, 10], you can use the definite integral. The definite integral represents the area bounded by the function and the x-axis within the given interval.
First, let's find the integral of f(x) with respect to x:
∫[4, 10] (10x - 5) dx
To integrate the function, you can use the power rule of integration. The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1), where n is not equal to -1.
Applying the power rule to our function, we get:
∫[4, 10] (10x - 5) dx = [(10/2)x^2 - 5x] evaluated from 4 to 10
Evaluating the integral at the upper and lower limits, we get:
[(10/2)(10^2) - 5(10)] - [(10/2)(4^2) - 5(4)]
Simplifying further:
[50(100) - 50] - [10(8) - 20]
= (5000 - 50) - (80 - 20)
= 4950 - 60
= 4890 square units
Therefore, the area of the region under the graph of the function f(x) on the interval [4, 10] is 4890 square units.