Answer:
The additional amount of area covered between 4 to 9 days is 23.71 cm2
Step-by-step explanation:
As the relation is given as a combination of two functions so integration by parts is carried out thus

In order to solve this integral, integration by parts is to be carried out which is given as

Where
u(x) is a function of x
v(x) is a function of x
u' is the derivative of u wrt to x
Also u and v are defined on using the following sequence ILATE RULE (Inverse, Logarithmic, Algebraic, Trigonometric, Exponent)
As here Logarithmic function is present which is taken as u and the algebraic function is taken as v so


Substituting the values in equation gives

Now solving the definite integral
![\int\limits^9_4 {\sqrt{t}\, ln t} \, dt=ln t (\frac{2}{3}t^{3/2}) -\frac{4}{9}(t^{3/2}) +C\\\int\limits^9_4 {\sqrt{t}\, ln t} \, dt=[ln t (\frac{2}{3}t^{3/2}) -\frac{4}{9}(t^{3/2})]_{9} -[ln t (\frac{2}{3}t^{3/2}) -\frac{4}{9}(t^{3/2})]_{4}\\\int\limits^9_4 {\sqrt{t}\, ln t} \, dt=18 ln (9)-\frac{16}{3} ln 4 -\frac{76}{9}\\A=\int\limits^9_4 {\sqrt{t}\, ln t} \, dt=23.71 cm^2\\](https://tex.z-dn.net/?f=%5Cint%5Climits%5E9_4%20%7B%5Csqrt%7Bt%7D%5C%2C%20%20ln%20t%7D%20%5C%2C%20dt%3Dln%20t%20%28%5Cfrac%7B2%7D%7B3%7Dt%5E%7B3%2F2%7D%29%20-%5Cfrac%7B4%7D%7B9%7D%28t%5E%7B3%2F2%7D%29%20%2BC%5C%5C%5Cint%5Climits%5E9_4%20%7B%5Csqrt%7Bt%7D%5C%2C%20%20ln%20t%7D%20%5C%2C%20dt%3D%5Bln%20t%20%28%5Cfrac%7B2%7D%7B3%7Dt%5E%7B3%2F2%7D%29%20-%5Cfrac%7B4%7D%7B9%7D%28t%5E%7B3%2F2%7D%29%5D_%7B9%7D%20-%5Bln%20t%20%28%5Cfrac%7B2%7D%7B3%7Dt%5E%7B3%2F2%7D%29%20-%5Cfrac%7B4%7D%7B9%7D%28t%5E%7B3%2F2%7D%29%5D_%7B4%7D%5C%5C%5Cint%5Climits%5E9_4%20%7B%5Csqrt%7Bt%7D%5C%2C%20%20ln%20t%7D%20%5C%2C%20dt%3D18%20ln%20%289%29-%5Cfrac%7B16%7D%7B3%7D%20ln%204%20-%5Cfrac%7B76%7D%7B9%7D%5C%5CA%3D%5Cint%5Climits%5E9_4%20%7B%5Csqrt%7Bt%7D%5C%2C%20%20ln%20t%7D%20%5C%2C%20dt%3D23.71%20cm%5E2%5C%5C)
So the additional amount of area covered between 4 to 9 days is 23.71 cm2