Zack K said:Homework Statement
Suppose that: 0∫2f(x)dx = 2 1∫2f(x)dx = -1 and 2∫4 = 7, find 0∫1f(x+1)dx
Homework Equations
a∫bf(x) = F(b) - F(a)
The Attempt at a Solution
So in these types of integration, we are needed to use u-substitution, the problem is, using u-substitution requires you to have the equation of the graph to get an exact value, but we just have f(x).I already evaluated 0∫1f(x)dx = 3 if that helps at all.
Without punctuation, he line above took me a while to figure out. At first I thought this meant ##\int_0^2 f(x)dx = 2 \cdot \int_1^2 f(x)dx = -1##. IOW that the values of the two integrals were -1 and -1/2, respectively.Zack K said:Homework Statement
Suppose that: 0∫2f(x)dx = 2 1∫2f(x)dx = -1 and 2∫4 = 7, find 0∫1f(x+1)dx
No, you don't need the equation of the function (f in this case). Just do a substitution and see what you get.Zack K said:Homework Equations
a∫bf(x) = F(b) - F(a)
The Attempt at a Solution
So in these types of integration, we are needed to use u-substitution, the problem is, using u-substitution requires you to have the equation of the graph to get an exact value, but we just have f(x).I already evaluated 0∫1f(x)dx = 3 if that helps at all.
The formula for finding the area of f(x+1) is the same as finding the area under a curve in calculus. It is given by the definite integral ∫f(x+1)dx, where f(x+1) is the function and dx represents the differential of x.
f(x+1) represents the function that is being integrated to find the area under the curve. It is a mathematical representation of the relationship between the input variable x and the output variable f(x+1).
Yes, the area of f(x+1) can be negative if the function f(x+1) dips below the x-axis. This indicates that the function has negative values for certain inputs, resulting in a negative area under the curve.
If the function f(x+1) is not given, you can still find the area by using the graphical interpretation of the definite integral. First, graph the function and then find the points where the function intersects the x-axis. Next, find the area of each individual shape formed by the x-axis and the function, and add them together to get the total area.
No, the only way to find the area of f(x+1) is by using integration. This is because the area under a curve is defined by the definite integral, and there is no other mathematical method to accurately calculate this area.