Question

Use the concept of the definite integral to find the total area between the graph off(x) and thex-axis, by taking the limit of the associated right Riemann sum. Write the exact answer. Do not round. (Hint: Extra care isneeded on those intervals wheref(x) < 0. Remember that the definite integral represents a signed area.)f(x) = 2x + 5 on [0, 4]

Answer 1

The given function is

`\begin{gathered} f(x)=2x+5\text{ on \lbrack{}0, 4\rbrack} \\ \end{gathered}`

The total area between the f(x) and x-axis is given by

`\begin{gathered} \int ^4_0f(x)dx=\int ^4_0(2x+5)dx \\ =\lbrack x^2+5x+c\rbrack^4_0 \\ =\lbrack4^2+5(4)+c\rbrack-\lbrack0^2+5(0)+c\rbrack \\ =\lbrack16+20+c\rbrack-c \\ =36\text{ square units} \end{gathered}`