Find the arc length of the graph of y = x from x = 0 to x = 3 using calculus.
∫ √[1 + (dx/dy)^2] dx (from a to b)
first, take the derivative of x and plug it into dx/dy, square that
then, add 1 and square root it
last, take the antiderivative
and of course, plug in the numbers
To find the arc length of the graph of y = x from x = 0 to x = 3, we can use calculus and the formula for arc length.
The formula for arc length of a function f(x) from a to b is given by:
L = ∫[a, b] √(1 + (f'(x))²) dx
In this case, our function is f(x) = x, and we want to find the arc length from x = 0 to x = 3.
Step 1: Find the derivative of f(x):
f'(x) = 1
Step 2: Substitute the values into the formula:
L = ∫[0, 3] √(1 + (1)²) dx
L = ∫[0, 3] √(1 + 1) dx
L = ∫[0, 3] √2 dx
Step 3: Integrate:
L = √2 ∫[0, 3] dx
L = √2 [x] from 0 to 3
L = √2 (3 - 0)
L = 3√2
Therefore, the arc length of the graph of y = x from x = 0 to x = 3 is 3√2.