find the area under the curve y=3log1.2x between x = 1.2 and x = 3
To find the area under the curve y = 3log₁.₂x between x = 1.2 and x = 3, we will use the definite integral.
Step 1: Determine the integral expression for the given function.
The integral expression for the given function is:
∫[1.2, 3] 3log₁.₂x dx
Step 2: Simplify the integral expression.
To simplify the integral expression, we can use a change of base formula to express the logarithm in terms of a common base like 10 or e.
Using the change of base formula, we get:
∫[1.2, 3] 3(log₁.₂x / log₁.₂10) dx
Step 3: Evaluate the integral.
Now we can evaluate the definite integral using the power rule for integration.
∫[1.2, 3] 3(log₁.₂x / log₁.₂10) dx
= [3 / log₁.₂10] ∫[1.2, 3] log₁.₂x dx
Applying the power rule for integration, we can rewrite the logarithmic function as:
= [3 / log₁.₂10] ∫[1.2, 3] ln(x) / ln(1.2) dx
Then, simplify the constant factor:
= C ∫[1.2, 3] ln(x) dx
Step 4: Evaluate the integral.
To evaluate the integral, we use the fundamental theorem of calculus.
∫ ln(x) dx = x(ln(x) - 1) + C
Therefore,
Area = C [3(ln(3) - 1) - 1.2(ln(1.2) - 1)]
By evaluating the expression, we can find the area under the curve y = 3log₁.₂x between x = 1.2 and x = 3.