Find the area under the curve of 
between 
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=4}
.
| Foundations:
|
| For solving the problem, we only require the use of the power rule for integration:
|
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int x^n dn = \frac{x^{n+1}}{n+1} + C.}
|
| Geometrically, we need to integrate the region between the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}
-axis, the curve, and the vertical lines Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 4}
.
|
Solution:
| Step 1:
|
| Set up the integral:
|
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_1^{\,4} \frac{8}{\sqrt{x}} \,dx.}
|
| Step 2:
|
| Using the power rule we have:
|
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\int_1^{\,4} \frac{8}{\sqrt{x}}\,dx} & = & \displaystyle {\int_1^{\,4} 8x^{-1/2}\,dx}\\ \\ & = & \displaystyle{\frac{8 x^{1/2}}{2} \Bigr|_{x=1}^4}\\ \\ & = & 4x^{1/2} \Bigr|_{x=1}^4. \end{array}}
|
| Step 3:
|
| Now we need to evaluate to get:
|
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4x^{1/2} \Bigr|_{x=1}^4\,=\,4\cdot 4^{1/2} - 4\cdot 1^{1/2} \,=\, 8 - 4 \,=\, 4.}
|
| Final Answer:
|
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_1^{\,4} \frac{8}{\sqrt{x}} \,dx\,=\,4.}
|
Return to Sample Exam