Evaluate the definite integral. \[ \int_{0}^{3} 10 x d x \]
Solution
Step 1 ：The integral of a function can be thought of as the area under the curve of the function. In this case, we are asked to find the definite integral of the function 10x from 0 to 3. This means we are finding the area under the curve of the function 10x from x=0 to x=3.
Step 2 ：The integral of 10x is \(5x^2\). To find the definite integral, we evaluate the antiderivative at the upper limit and subtract the antiderivative evaluated at the lower limit.
Step 3 ：Substitute x=3 into the antiderivative function \(5x^2\), we get \(5*3^2=45\).
Step 4 ：Substitute x=0 into the antiderivative function \(5x^2\), we get \(5*0^2=0\).
Step 5 ：Subtract the value of the antiderivative at the lower limit from the value at the upper limit, we get \(45-0=45\).
Step 6 ：Final Answer: The definite integral of the function 10x from 0 to 3 is \(\boxed{45}\).
