Find the area under the curve of 
between the 
-axis and 
.
| Foundations:
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| For solving the problem, we only require the use of the power rule for integration:
|
 for  ,
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| For setup of the problem we need to integrate the region between the x - axis, the curve, 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 = 0}
(the y-axis), 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 = 2}
.
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Solution:
| Step 1:
|
| Set up the integral:
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- 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_0^{2} 6x^2 + 2x \,dx.}
|
| Step 2:
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| Using the power rule we have:
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- 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 _0^2 6x^2+2x \,dx \,=\, 6\cdot \frac{x^3}{3}+2\cdot \frac{x^2}{2} \Bigr|_{x\,=\,0}^2\,=\,2x^3+x^2 \Bigr|_{x\,=\,0}^2. }
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| Step 3:
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| Finally, we need to evaluate:
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- 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 2x^3 + x^2 \Bigr|_{x\,=\,0}^2 = (2(2)^3+(2)^2)-(0+0) = 20.}
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| Final Answer:
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- 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_0^{\,2} 6x^2 + 2x \,dx\,=\,20.}
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