Final answer:
To calculate the area bounded by the given curve and lines, integrate the function y = x^2 + 3 from x = -2 to x = 3. Compute the antiderivative, evaluate at the limits, and find the difference to get the area.
Step-by-step explanation:
To find the area bounded by the curve y = x^2 + 3, the lines x = -2, x = 3, and the x-axis, we integrate the function y = x^2 + 3 from x = -2 to x = 3. This is a problem of calculating the definite integral of the function which will provide the total area under the curve within the specified limits.
The integral we need to evaluate is ∫ (x^2 + 3) dx from x = -2 to x = 3. To do this, we first compute the antiderivative of the function, which is (1/3)x^3 + 3x. We then evaluate this antiderivative at the upper and lower limits of integration and subtract the lower limit value from the upper limit value.
The definite integral is therefore calculated as follows:
[¼x^3 + 3x]_{-2}^{3} = (¼·3^3 + 3·3) - (¼·(-2)^3 + 3·(-2))
After performing the calculations, we find the exact numerical value for the area which represents the space bounded by the curve and the given lines along with the x-axis.