I know that in order to get the Flux of a star (or something else) in a particular filter from its SED (luminosity per unit wavelength), I need to convolve the spectrum (SED) with the filter response. Most of the formulas I see to do this are $$F_{b}=\dfrac{\int f(\lambda) T_{b}(\lambda)d\lambda}{\int T_{b}(\lambda)d\lambda}\,,$$ where $f(\lambda)$ is the SED, and $T_b$ is the filter response in band $b$.
This formula seems very different from a mathematical convolution that I would write as $$f*g(\lambda)=\int_{-\infty}^{\infty} f(x) g(\lambda-x)dx\,.$$
Are these two convolutions the same thing? Or is the "astronomy convolution" a different thing (e.g. SED Fitter python package)?