Let f be a continuous function on the interval [0,2] which satisfies integral of f(x)dx=5 when b is 2 and a is 0. Give this information compute the definite integral f(2y)dy when b is 1 and a is 0. Solve this
NanakXx
To solve the given integral ∫[0,1] f(2y)dy, we can use the substitution method. Let's break it down step by step:
Step 1: Perform the substitution.
Let's substitute u = 2y, which implies that du = 2dy. Rearranging the equation, we get dy = (1/2)du.
Step 2: Adjust the limits of integration.
When y = 0, u = 2(0) = 0. And when y = 1, u = 2(1) = 2. So the new limits of integration will be u = 0 to u = 2.
Step 3: Rewrite the integral.
Using the substitution from Step 1, we can rewrite the given integral as follows:
∫[0,1] f(2y)dy = ∫[0,2] f(u) * (1/2)du
Step 4: Solve the integral.
Now, we can solve the integral ∫[0,2] f(u) * (1/2)du. According to the information given, ∫[0,2] f(x)dx = 5. Since f(x) is a continuous function, we can use the Fundamental Theorem of Calculus to find the value of this integral.
∫[0,2] f(x)dx = F(x) |[0,2]
Since ∫[0,2] f(x)dx = 5, we have:
F(2) - F(0) = 5
Step 5: Substitute back the variable u.
Replace x with u in the equation above, we get:
F(u) - F(0) = 5
Now, we need to substitute back the variables to find the integral ∫[0,2] f(u) * (1/2)du.
Using the equation from Step 5, we have:
(F(u) - F(0)) * (1/2) = 5/2
Therefore, the value of the integral ∫[0,1] f(2y)dy is 5/2.