To approximate the area between the graphs of [tex]\( y = \ln(x) \)[/tex] and [tex]\( y = 7x e^x \)[/tex] over the interval [tex]\([1, 2]\)[/tex], follow these steps:
1. Understand the functions and the interval:
- The first function is [tex]\( y = \ln(x) \)[/tex].
- The second function is [tex]\( y = 7x e^x \)[/tex].
- The interval over which we are calculating the area is [tex]\([1, 2]\)[/tex].
2. Calculate the definite integrals of each function over the interval [tex]\([1, 2]\)[/tex]:
- The integral of [tex]\( y = \ln(x) \)[/tex] over [tex]\([1, 2]\)[/tex].
- The integral of [tex]\( y = 7x e^x \)[/tex] over [tex]\([1, 2]\)[/tex].
3. Definite integral of [tex]\( y = \ln(x) \)[/tex] over [tex]\([1, 2]\)[/tex]:
[tex]\[
\int_{1}^{2} \ln(x) \, dx \approx 0.3863
\][/tex]
4. Definite integral of [tex]\( y = 7x e^x \)[/tex] over [tex]\([1, 2]\)[/tex]:
[tex]\[
\int_{1}^{2} 7x e^x \, dx \approx 51.7234
\][/tex]
5. Find the absolute difference between the two areas:
[tex]\[
\left| 51.7234 - 0.3863 \right| = 51.3371
\][/tex]
6. Round the result to the nearest hundredth:
[tex]\[
51.34
\][/tex]
Therefore, the approximate area between the graphs of [tex]\( y = \ln(x) \)[/tex] and [tex]\( y = 7x e^x \)[/tex] over the interval [tex]\([1, 2]\)[/tex] is [tex]\( \boxed{51.34} \)[/tex].
