Sorry I posted this question earlier with a error.
Can anyone help me solve this now?
Can someone help me answer this?
If a < 5 the define integral [a, 4] 2.4e^(1.4x)dx = 44
Find the value a
Define integral = integral sign
a = lower limit
5 = upper limit
To solve this problem, we need to find the value of "a" that makes the given definite integral equal to 44. Here's how we can approach it step by step.
Step 1: Set up the definite integral.
The given definite integral is:
∫[a,4] 2.4e^(1.4x)dx = 44
Step 2: Evaluate the definite integral.
To evaluate this integral, we need to find the antiderivative of the function 2.4e^(1.4x). The antiderivative of e^(kx) is e^(kx) / k. Therefore, the antiderivative of 2.4e^(1.4x) is (2.4/1.4)e^(1.4x).
By applying the Fundamental Theorem of Calculus, we can evaluate the definite integral by subtracting the antiderivative evaluated at the upper limit (4) from the antiderivative evaluated at the lower limit (a).
[ (2.4/1.4)e^(1.4x) ] from a to 4 = 44
Step 3: Calculate the definite integral.
Now we can perform the calculation by plugging in the upper and lower limits and setting it equal to 44:
(2.4/1.4)e^(1.4*4) - (2.4/1.4)e^(1.4*a) = 44
Step 4: Solve for "a".
To solve for "a", we isolate the variable on one side of the equation. Rearrange the equation by moving the second term to the other side:
(2.4/1.4)e^(1.4*4) = 44 + (2.4/1.4)e^(1.4*a)
Step 5: Simplify and solve.
Now you can simplify the equation algebraically and solve for "a". Let's denote the value (2.4/1.4)e^(1.4*4) as "B" for simplicity:
B = 44 + (2.4/1.4)e^(1.4*a)
From here, you can subtract 44 from both sides of the equation:
B - 44 = (2.4/1.4)e^(1.4*a)
Then, divide both sides of the equation by (2.4/1.4):
(B - 44)/(2.4/1.4) = e^(1.4*a)
Finally, to get rid of the exponent, take the natural logarithm (ln) of both sides:
ln[(B - 44)/(2.4/1.4)] = 1.4*a
Now we can solve for "a" by dividing both sides of the equation by 1.4:
a = ln[(B - 44)/(2.4/1.4)] / 1.4
By substituting the value of "B" and simplifying the expression, you can find the value of "a".